Chaotic Communication System with Modulation of Nonlinear Elements

ABSTRACT

A chaotic communication system employs transmitting and receiving chaotic oscillating circuits. One improvement to first-generation systems is the ability to modulate a nonreactive element in the transmitting circuit, thus increasing modulation bandwidth. Other features include insertion of a gain control amplifier in a chaotic receiver; signal filtering in chaotic transmitters and receivers; use of chaotic modulation techniques for cellular telephony applications; dual-transmitter and receiver systems; a dual receiver synchronization detector; interfaces to communication systems; analog chaotic signal modulation; use of multiple chaotic transmitters and receivers; digital algorithm improvement using a cube-law nonlinear component; a Gb-only receiver; a Gb-only transmitter; and positive slope transmitter and receiver systems.

This application is a continuation of commonly-owned, co-pending U.S.application Ser. No. 11/270,502 filed on Nov. 2, 2005, which is acontinuation of Ser. No. 09/317,124 filed on May 24, 1999. U.S.application Ser. No. 09/317,124, granted as U.S. Pat. No. 6,980,656issued on Dec. 27, 2005, is a continuation-in-part of U.S. applicationSer. No. 09/116,661, filed on Jul. 17, 1998, now abandoned. Theabove-cited applications are incorporated into the body of thisapplication in their entireties.

TECHNICAL FIELD

This invention relates generally to information transmission techniquesinvolving modulation and demodulation of a chaotic carrier signal. Manyaspects of the invention involve transmitting information by modulatingvarious characteristics of nonreactive circuit elements of a chaotictransmitter. The invention has broad application to communicationssystems, radar systems and other systems that transmit and receiveinformation over wire, radio frequencies, light (including fiber optic)and acoustic channels.

BACKGROUND OF THE INVENTION

Techniques for modulating carrier signals in order to transmitinformation between two points are well known. In systems employingfrequency modulation, for example, a carrier signal is modulated bychanging the frequency of the signal in accordance with an informationsignal such as a human voice. Amplitude-modulated systems change theamplitude of a fixed-frequency signal in accordance with an informationsignal. Other modulation techniques have been developed over the yearsto optimize transmission characteristics, to optimize signal bandwidth,and to overcome noisy transmission environments.

So-called “chaotic” signals provide a particularly interesting, simple,and useful means of modulating information signals in a manner that canincrease noise immunity and reduce the power levels needed to transmitinformation. As explained in the aforementioned application, which isbodily incorporated herein, these signals can be modulated in variousways to transmit information. The modulation bandwidth available whenusing such techniques, however, has been determined to be generallylimited to 10 to 15% of the tank circuit frequency in the transmittingcircuit. This limitation is believed to be due to the fact that changinglump parameters in the transmitter causes a certain amount of settlingtime before the receiver can synchronize with the changed transmitterparameters.

The present inventors have discovered a technique for modulating thetransmitting signal in a manner that results in much faster signalstability, thus reducing the amount of time required to synchronize thereceiver and increasing the modulation bandwidth dramatically. Otherfeatures and advantages provided by the present invention will becomeapparent upon reading this specification in conjunction with thefigures.

The following description begins by reviewing the subject matter of theaforementioned application as a departure point for explaining theprinciples of the present invention. Circuits, principles andembodiments described in the aforementioned application will be referredgenerally to as “first-generation,” while those newly presented in thisapplication will be referred to generally as “second-generation” or“improved.” These labels are not intended in any way to be limiting.Moreover, many of the second-generation circuits and principles can beused in conjunction with first-generation circuits and vice versa.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a diagram of a Chua circuit according to the prior art.

FIG. 1B is a communications system according to the prior art.

FIG. 1C is a diagram indicating the resistance—voltage characteristic ofa non-linear resistor used in a Chua circuit.

FIG. 1D is a diagram of the operating regimes of a Chua circuit mappedinto a lump parameter plane.

FIG. 2A is a schematic of a transmitter of a first-generation systemwherein a capacitor 237 is switched to modulate a chaotic signal.

FIG. 2B shows a conventional receiver that can be used with thetransmitter of FIG. 2A.

FIG. 3A shows another embodiment of a first-generation transmitter thatproduces a vocabulary of chaotic signals.

FIG. 3B shows a receiver for use with the embodiments of FIG. 3A andFIG. 4A.

FIG. 4A shows another embodiment of a first-generation transmitter thatproduces a vocabulary of chaotic signals in which all such signals canbe mapped to the same combinations of the lump parameter plane of FIG.1D.

FIG. 4B shows another embodiment of a first-generation receiver for usewith the embodiments of FIG. 3A and FIG. 4A, this receiver being usablewith a simple counter circuit for determining a beat frequency.

FIG. 4C shows another receiver for use with the embodiment of FIG. 3Aand FIG. 4B using a synchronizing resistor 385.

FIG. 4D shows another receiver usable with a simple counter circuit fordetermining a beat frequency using a synchronizing resistor formed froma combination of resistors.

FIG. 4E shows another receiver similar to that of FIG. 4D, but whichadds an emitter follower 353 to isolate the oscillator portion 361 frompoint 287.

FIG. 4F shows a receiver including a simple counter circuit fordetermining a beat frequency, wherein resistors provide a synchronizingelement to lock the incoming voltage of the communications channel andthe receiver generated voltage. In this embodiment, voltage follower 363isolates the receiver generated signal from the incoming signal andallows the receiver generated signal to feedback into the oscillatorportion of the Chua circuit to cause faster synchronization.

FIG. 5 shows a generalized communications system with a synchronizingfilter 550 according to a first-generation embodiment of the invention.

FIG. 6A shows a conventional transmitter with a Kennedy non-linear diodethat can be modulated according to the principles of a second-generationsystem.

FIG. 6B shows a conventional transmitter with a Caltech non-linear diodethat can be modulated according to the principles of a second-generationsystem.

FIG. 6C shows a transmitter with a novel non-linear diode that can bemodulated in accordance with the principles of a second-generationsystem.

FIG. 7A shows how a switch 735 c can be used to modulate voltage-currentcharacteristics of a nonlinear diode in accordance with asecond-generation embodiment.

FIG. 7B shows generally how an information signal 736 can be used tomake and break a switch 735 c (or other switch-like device) to modulatea negative resistance in accordance with a second-generation embodiment.

FIG. 7C shows the effects of changing various resistive values in aCaltech diode (FIG. 6B) and Kennedy diode (FIG. 6A) on slope.

FIG. 8 shows how the slope of the current-voltage characteristic curvefor a nonlinear element can be changed in order to change the rotationof a strange attractor phase plane according to a second-generationembodiment. FIG. 9A shows modulation limits for Ga and Gb according to asecond-generation transmitter.

FIG. 9B maps current-voltage characteristic curves between a modulatedtransmitter and a receiver.

FIG. 9C shows how positive breakpoints can be established for anonlinear diode according to various second-generation embodiments.

FIG. 10 shows a field effect transistor 1001 placed across diodes 652and 655 to implement on-off keying according to a second-generationembodiment.

FIG. 11 shows a gain control amplifier 1146 inserted into a receivingcircuit that reduces or eliminates the need for an automatic gaincontrol (AGC) amplifier on the input at point 1191.

FIG. 12 shows a general case of a gain control amplifier 1202 embeddedin a receiving circuit.

FIG. 13 shows a transmitter including an operational amplifier 1308 thatisolates a filter 1309 from the chaotic subsystem 222 according to asecond-generation embodiment of the invention.

FIG. 14A shows low-pass filtering characteristics of a transmitteraccording to a second-generation embodiment of the invention.

FIG. 14B shows bandpass filtering characteristics of a transmitteraccording to a second-generation embodiment of the invention.

FIG. 15 shows a receiver including noise filters 1593, 1594 and 1597 tofilter out noise components introduced by the transmission channelaccording to a second-generation embodiment of the invention.

FIG. 16 shows a receiver including noise filters 1666, 1694 and 1697 tofilter out noise components introduced by the transmission channelaccording to a second-generation embodiment of the invention.

FIG. 17 shows a receiver including filters 1793, 1766, 1794, 1797 and1721, wherein filter 1721 works with automatic gain control amplifier1746 to further reduce noise generated in subsystem 1726.

FIG. 18A shows a cell phone system incorporating a baseband modem usingchaotic modulation principles according to a second-generationembodiment of the invention.

FIG. 18B shows a cell phone system using chaotic modulation principlesat the intermediate frequency level according to a second-generationembodiment of the invention.

FIG. 18C shows a cell phone system using chaotic modulation principlesat the radio frequency level according to a second-generation embodimentof the invention.

FIG. 19A shows a second-generation system employing various principlesof the present invention, including a transmitter with a nonlinearelement 1904 modulated with an information signal, and a receiverincluding a nonlinear element 1905 and a synchronizing resistor Rsync.

FIG. 19B shows a current/voltage curve 1910 for a nonlinear diodesuperimposed over a load line 1911.

FIG. 19C shows two “single-scroll” attractors 1920 and 1930 orbitingaround equilibrium points at the intersection of a nonlinear diodecurrent-voltage characteristic curve and a load line, illustrating a DCanalysis of a transmitter.

FIG. 19D shows how the current-voltage characteristic curve can bechanged at the transmitter to move the equilibrium points between threedifferent positions while slope Ga and the breakpoint positions are heldconstant.

FIG. 19E shows a strange attractor moving with the equilibrium point asa nonlinear circuit element is modulated with an information signal,causing changes in the slope of part of the characteristic curve.

FIG. 19F shows one technique for changing a nonlinear diodecurrent-voltage characteristic curve using an ideal switch SW1 and aresistor in series with the ideal switch that is also in parallel withone of the nonlinear diode resistors.

FIG. 19G shows the result of modulation on the voltage across thenonlinear diode.

FIG. 20A shows a voltage (V1) versus voltage (V2) versus time (T) plotof a chaotic signal (double scroll strange attractor) withoutmodulation.

FIG. 20B shows the plot of FIG. 20A when modulated with an informationsignal.

FIG. 20C shows a voltage (V2) versus current (I3) versus time (T) plotof a chaotic signal (double scroll strange attractor) withoutmodulation.

FIG. 20D shows the plot of FIG. 20C when modulated with an informationsignal.

FIG. 20E shows a voltage (V1) versus current (I3) versus time (T) plotof a chaotic signal (single scroll attractor) without modulation.

FIG. 20F shows the plot of FIG. 20E when modulated with an informationsignal (single scroll modulation).

FIG. 21A shows a nonlinear diode current-voltage characteristic curvewhere resistor R1 is set to 1200 ohms (double scroll attractor).

FIG. 21B shows a nonlinear diode current-voltage characteristic curvewhere resistor R1 is set to 1210 ohms (double scroll attractor).

FIG. 21C shows a nonlinear diode current-voltage characteristic curvewhere resistor R1 is set to 1220 ohms (double scroll attractor).

FIG. 21D shows a voltage-current phase space map (V1 vs. V2 vs. I3)corresponding to the nonlinear diode curve of FIG. 21A.

FIG. 21E shows a voltage-current phase space map (V1 vs. V2 vs. I3)corresponding to the nonlinear diode curve of FIG. 21B. As compared toFIG. 21D, the strange attractor on the left side is “squashed.”

FIG. 21F shows a voltage-current phase space map (V1 vs. V2 vs. I3)corresponding to the nonlinear diode curve of FIG. 21C. As compared toFIG. 21E, the strange attractor on the left side is even more“squashed.”

FIG. 21G shows a frequency plot corresponding to the nonlinear diodecurve of FIG. 21A.

FIG. 21H shows a frequency plot corresponding to the nonlinear diodecurve of FIG. 21B.

FIG. 21I shows a frequency plot corresponding to the nonlinear diodecurve of FIG. 21C.

FIG. 22A shows a voltage (V1) versus voltage (V2) versus time (T) plotof a chaotic signal (single scroll strange attractor) withoutmodulation.

FIG. 22B shows the plot of FIG. 22A when modulated with an informationsignal.

FIG. 22C shows a voltage (V2) versus current (I3) versus time (T) plotof a chaotic signal (single scroll strange attractor) withoutmodulation.

FIG. 22D shows the plot of FIG. 22C when modulated with an informationsignal.

FIG. 22E shows a voltage (V1) versus current (I3) versus time (T) plotof a chaotic signal (single scroll strange attractor) withoutmodulation.

FIG. 22F shows the plot of FIG. 22E when modulated with an informationsignal.

FIG. 23A shows a nonlinear diode current-voltage characteristic curvewhere resistor R1 is set to 1930 ohms (single scroll attractor).

FIG. 23B shows a nonlinear diode current-voltage characteristic curvewhere resistor R1 is set to 1940 ohms (single scroll attractor).

FIG. 23C shows a nonlinear diode current-voltage characteristic curvewhere resistor R1 is set to 1950 ohms (single scroll attractor).

FIG. 23D shows a voltage-current phase space map (V1 vs. V2 vs. I3)corresponding to the nonlinear diode curve of FIG. 23A.

FIG. 23E shows a voltage-current phase space map (V1 vs. V2 vs. I3)corresponding to the nonlinear diode curve of FIG. 23B. As compared toFIG. 23D, the strange attractor is elongated.

FIG. 23F shows a voltage-current phase space map (V1 vs. V2 vs. I3)corresponding to the nonlinear diode curve of FIG. 23C. As compared toFIG. 23E, the strange attractor is even more elongated.

FIG. 23G shows a frequency plot corresponding to the nonlinear diodecurve of FIG. 23A.

FIG. 23H shows a frequency plot corresponding to the nonlinear diodecurve of FIG. 23B.

FIG. 23I shows a frequency plot corresponding to the nonlinear diodecurve of FIG. 23C

FIG. 24 shows a dual-transmitter configuration (1200, 1205) according toa second-generation embodiment of the invention.

FIG. 25 shows a dual-receiver configuration (601, 1370) according to asecond-generation embodiment of the invention.

FIG. 26 shows a subtraction circuit for detecting a voltage differenceacross various points (e.g., point 1450) in FIG. 25.

FIG. 27 shows an absolute value circuit that can be used in conjunctionwith a detector function.

FIG. 28 shows a dual receiver synchronization detector circuit accordingto a second-generation embodiment of the invention.

FIG. 29 shows a dual receiver synchronization detector circuit in whichsignals are subtracted, absolute valued and then subtracted.

FIG. 30 shows a detector circuit for detecting voltage changes betweenpoints 287 and 1415 or points 1440 and 1470 in FIG. 25.

FIG. 31 shows a baseband transmitter interface circuit for interfacing amodulated chaotic transmitter to a communication system.

FIG. 32 shows a baseband receiver interface circuit for interfacing areceiver to a communication system.

FIG. 33 shows a system in which a chaotic transmitter and receiver areinterfaced to an infrared amplitude modulated subsystem 3090.

FIG. 34 shows how a chaotic transmission and reception system can beinterfaced to a radio transmitter/receiver pair 3100 and 3110.

FIG. 35 shows a balanced cable driver circuit 2455 that can be used topass a chaotic signal over a twisted pair or coaxial cable system.

FIG. 36 shows a balanced cable receiver circuit 2880 that can be used tointerface a chaotic receiver 2860 to a twisted pair or coaxial cablesystem.

FIG. 37A shows curves representing chaotic operating regions fordifferent values of a synchronizing resistor 660 for a Caltech diodeimplementation (FIG. 6B).

FIG. 37B shows curves representing chaotic operating regions fordifferent values of a synchronizing resistor 608 for a Kennedy diodeimplementation (FIG. 6A).

FIG. 37C shows what happens when the transmitter capacitor 215 (FIG. 6B)is varied and the receiver capacitors 355 and 1490 (FIG. 25) are set tofixed values with the nonlinear diode characteristic curve set at afixed value.

FIG. 38 shows a technique for doubling a signal rate using fourunmodulated oscillator/transmitters and corresponding receivers.

FIG. 39 shows a technique for increasing the digital signal transmissionrate using multiple chaotic transmitters and matched receivers.

FIG. 40 shows how a nonlinear circuit can be replaced with two functionsthat represent only the Gb slopes, referred to as a “Gb-only”transmitter or receiver.

FIG. 41 shows a detector design in which a nonlinear diode is replacedwith +/−Gb slope detectors (5350 and 5340).

FIG. 42 shows a dual receiver design using sample-and-hold circuits withoutputs 5380, 5360, and 5370.

FIG. 43 shows a transmitter that modulates only the slope Gb.

FIG. 44 shows a dual-transmitter system that modulates only the slopeGb.

FIG. 45A shows a dual receiver design wherein a nonlinear diode isreplaced with a +/−Gb detector and voltage Vb in a negative resistorcircuit.

FIG. 45B shows a dual receiver design that is a variation on that ofFIG. 45A.

FIG. 46 shows a current-voltage characteristic curve for certainembodiments of the invention that modulate and detect a positive slope.

FIG. 47 shows a receiver in which a nonlinear diode is replaced with aGb+ detector.

FIG. 48 shows a Gb+ dual receiver design including a sample and holdcircuit.

FIG. 49 shows a Gb+ only transmitter using Gb+ slope modulation andvoltage modulation.

FIG. 50 shows a digital to analog Gb+ only transmitter.

FIG. 51 shows a current-voltage characteristic curve for certainembodiments of the invention.

FIG. 52 shows a current-voltage characteristic curve for a positive Gbvoltage current M-ary modulation system.

I. FIRST-GENERATION EMBODIMENTS AND TECHNIQUES

Referring to FIG. 1A, a circuit 1 known as a “Chua” circuit oscillateschaotically. The term “chaos” applies to dynamic systems that followsimple dynamical rules, but whose state function trajectory is sosensitive to the system's initial conditions that its state after anarbitrary time-period cannot, in practical terms, be predicted. That is,its state could be predicted if it were possible to model the systemwith an arbitrary degree of precision.

Chaotic systems evolve deterministically, and their chaotic state pathsare cyclic, but very complex and with extremely long cycle-lengths. Inreal systems, however, with extremely long cycle periods, it may be oflittle practical significance that their behavior is cyclical becausethe physical systems that generate the behavior may not be sufficientlystable for the system to ever return to the same dynamical system in itssame initial state. For example, the component values of an electricalcircuit may not remain precisely constant for 600 years.

The Chua circuit is a simple electrical circuit that exhibits chaoticbehavior. It has been studied extensively and used to demonstrate manyof the chaotic patterns observed in many physical systems. Referring nowalso to FIG. 1A, the basic Chua circuit includes a non-linear resistanceelement 10, characterized by a non-linear voltage-current characteristiccurve. In a typical configuration, the curve is piece-wise linear withsymmetrical slope discontinuities around the zero-axis. That is:I_(R)=G_(a)V_(R)+(½)(G_(a)−G_(b)){|v_(R)+B_(p)|−|v_(R)−B_(p)|} whereG_(a) and G_(b) are the slopes of respective linear portions of thepiecewise-linear current/voltage curve characterizing the non-linearresistor and B_(P) is the absolute value of the two voltage points atwhich the discontinuities in the current/voltage curve lie as shown inFIG. 1C. The circuit has a circuit-driving subsystem 2 (e.g., an L-Ctank circuit), and a response subsystem 3, which includes for example acapacitance C1 and non-linear resistor 10, wherein the two systems areinterconnected through a resistor 25.

Referring to FIG. 1D, a given choice of values of the physicalcharacteristics of the components of the Chua circuit each correspond toa unique operating regime, some values of which may coincide with achaotic behavior of the Chua circuit. The operating regime may be mappedonto a coordinate system whose axes are the lump parameters, α=C₂/C₁ andβ=R²C₂/L. By choosing values of R (25), L (30), C₁ (15) and C₂ (20) sothat α and β lie in, for example, a double scroll region 60, a Chuacircuit can be made that will oscillate chaotically orquasi-periodically. Any point on the plot corresponds to a differentoperating behavior and a selected point does not exhaustively define aparticular path of state trajectories. A selected point on the curvescan correspond to radically different behaviors depending on the initialconditions.

Given a specified physical configuration and a specified initial statespecified by V₁, V₂, and I_(L), the voltages across C₁ (15) and C₂ (20)and the current through L (30), the evolution of the Chua circuit'sstate is deterministic, but chaotic. That is, any Chua circuit with thesame physical parameters and initial conditions will follow the samecourse of states over time and this course will repeat itself over avery long interval (perhaps many years). However, to an observer, thevalue of (for example) voltage V1 over a period of time shorter thanthis long interval looks like noise. Also, initial states that differonly slightly can follow widely different state paths. In addition, itspower spectral density function is spread over a wide range offrequencies, with a peak at the frequency of the fundamental of the L-Ctank circuit formed by L and C₂. However, compared to oscillators, suchas used to generate carriers for radio transmission, the peak is notpronounced; that is, it is very short and wide.

The Chua circuit, aside from being a classic device for demonstrating,studying, and modeling chaotic real-world systems, has also beenproposed as a basis for chaotic signal transmission. Generally atransmitting nonlinear dynamic circuit produces a chaotic signal thatcan be used to induce a receiving chaotic system to synchronize with it.The parameter of the transmitting chaotic circuit can be modulated orperturbed responsively to an information signal. The parameter can be ascalar, such as a voltage, tapped from the transmitting circuit and usedas a signal. The signal is applied to the receiving system, causing thereceiving system to synchronize with the transmitted signal. The chaoticsignal from the synchronized receiving circuit can be used with themodulated transmitted signal to recover the information signal accordingto various prior art schemes. The chaotic signals that can be derivedfrom an oscillating Chua circuit are similar to spread-spectrum signalsincluding a range of frequencies. Chua circuits have been made togenerate communications signals in frequency bands ranging from audio toradio frequency and in various media.

Various modulation schemes have been proposed. For example, a simplesignal summing system adds the information signal to the chaotic scalar.A more complex correlation system uses a signal divider and multiplierat the transmitter and receiver, respectively. In FIG. 1B, a prior artsystem uses a Chua circuit to transmit signals and receive signals. Thesystem has a transmitting Chua circuit 100 and an identical (in terms ofits chaotic oscillating properties) receiving Chua circuit 101. Thetransmitting Chua circuit 100 oscillates in a chaotic or semiperiodicregime.

Generally, the two chaotic circuits 100 and 101 can be synchronized bydriving a portion of the receiving chaotic oscillator 101 with a drivingfunction tapped from the transmitting chaotic oscillator 100. L-C tankcircuit 105 of the transmitting Chua circuit 100 is linked through aresistor 81 to the capacitor/non-linear resistor portion 106. The latterportion causes the oscillations of the L-C tank circuit to becomechaotic for certain values of the inductor 74, capacitors 71 and 73, andresistor 81 as discussed above with reference to FIG. 1D. The chaoticportion 108 of the identical receiving circuit 101, also acapacitor/non-linear resistor circuit, reproduces the driving signal.That is, the transmitting 100 and receiving 101 circuits followprecisely the same chaotic course of states (assuming no modulation istaking place in the transmitting circuit 100).

It is known that the transmitting 100 and receiving 101 circuits willremain synchronized even when a substantial amount of noise and/orinformation is injected into the driving signal. Thus, in the prior artembodiment of FIG. 1B, a signal current I_(i)(t) is injected by a driver76 that converts a signal voltage through an invertable coding functionc(v_(s)(t)). The decoded signal at the receiver is then obtained fromthe received current signal I_(d)(t) by applying the inverse codingoperation to the received current signal I_(d)(t) to obtain a voltagesignal containing the information signal.

Note that the term, “synchronous,” in this context, characterizes theconvergence of two state variables toward identical or linearly related,but continuously changing, sets of values. That is, a change in onevariable corresponds to a change in a synchronized variable that islinearly related to the change in the one variable. Thus, plotting onevariable against the synchronized variable over time, the result,theoretically, is a straight line. Synchronization of non-linearsystems, and the mathematical modeling of such systems, is described insome detail in U.S. Pat. Nos. 5,245,660; 5,473,694; 5,402,334;5,379,346; 5,655,022; 5,432,697; and 5,291,555, the entirety of each ofwhich is incorporated by reference herein.

Prior art systems have been discussed widely, but few practical workingdesigns are known. The problems with practical synchronization systemsare summarized in the introduction of U.S. Pat. No. 5,680,462.Synchronization systems are inherently noisy and error prone due, atleast in part, to the time it takes for synchronization to occur in anoisy channel and because noise induces state transitions in thereceiver since it causes a breakpoint to be crossed. For example, when atransmitting circuit is perturbed to encode a piece of information (abit), it takes a finite amount of time for the receiving circuit tobegin to follow the trajectory of the transmitted signal. Also,according to the prior art, modulation cannot span too great a range.Otherwise, a tightly locked synchronization, which is, according to theprior art, essential, cannot be maintained. In addition, the practicalproblems attending achievement of high data throughput, the providing ofreliable locking performance, and various purely practical designconsiderations have not received a great deal of attention. These priorart problems are addressed by the present invention in both thefirst-generation and second-generation embodiments.

According to one aspect of a first-generation system, the inventionprovides a spread-spectrum-like communications system that transmitsinformation in a chaotic signal. Other aspects of the invention include:

(a) a method for modulating a chaotic process to generate a signal toencode information in the signal;

(b) a method for modulating a circuit that generates a chaotic signal ina stable manner.

(c) transmitting and receiving chaotic circuits that are characterizedby rapid synchronization;

(d) a mechanism for imprinting and extracting information from twochaotic devices synchronized by a chaotic signal in which theinformation is embedded such that the chaotic signal can serve as theinformation carrier signal over a communication channel;

(e) a communications system that permits the modulation of a chaoticprocess so as to encode multiple independent streams of data on a samechaotic carrier signal, in effect, implementing an N-word vocabulary,where N corresponds to a number of stable chaotic oscillation statesthat are induced in a transmitter by modification of a property of atleast one of a resistance, a capacitance, and an inductance to tune anoscillating circuit of the transmitter.

Briefly, an embodiment of the invention employs a transmittingoscillating circuit capable of chaotic or quasiperiodic oscillation togenerate a (chaotic or quasiperiodic) carrier, preferably a voltagetapped through a voltage follower. A property of the transmittingoscillating circuit, in an embodiment, an auxiliary capacitance, isswitched on and off to vary the capacitance of an L-C tank portion of aChua oscillator. The switching is controlled by an information signal togenerate a modulated chaotic signal.

Switching is performed with an optical isolator that requires zerooutput biasing and introduces essentially no capacitance into thecircuit. This prevents any effect on the chaotic or quasi-periodicoperating regime of the circuit. An autonomous portion of a receivingoscillating circuit, substantially identical in terms of its oscillatingproperties, is driven by the modulated carrier. This establishes asynchronized chaotic or quasiperiodic oscillation in the receivingcircuit. A comparator is used to output the difference between thedriving modulated carrier and a synchronized signal tapped from thereceiving oscillator at a point corresponding to the transmittingcircuit tap used to generate the modulated carrier. This output providesthe recovered information signal.

In one embodiment, various elements of the transmitting chaotic circuitare switchably varied to maintain a constant operating regime so thatstrange attractors, with frequencies covering a wide selectable range,are generated. This is used to form a vocabulary of strange attractors.The frequency can be determined by the receiver in a very simple way bycounting pulses formed from a difference between the base signal in thereceiving circuit and the received signal. The difference in thefrequencies of the signal being transmitted and the base signalgenerated by the receiver indicates the “word” transmitted. In this way,if the vocabulary consists of N distinguishable oscillating frequencies,then log₂(N) bits can be transmitted with each modulating cycle.

According to one embodiment, the invention provides a communicationsdevice with a transmitting chaotic circuit. The transmitter has at leastone circuit element, the value of which affects a chaotic electricalproperty of the chaotic circuit. That is, a change in the magnitude ofthe circuit element changes the oscillating behavior of the chaotictransmitting circuit. The circuit element has multiple componentelements, at least one of which is isolated from the chaotic circuit bya switch. The configuration is such that when the switch is switched toa first state, the magnitude has a first value and when the switch isswitched to a second state, the magnitude of the component has a secondvalue. This causes the transmitter to oscillate over multipleoscillating regimes each corresponding to one of the values. The chaoticproperty can be applied to a communications channel to be picked up by areceiver.

Switching the circuit element allows, essentially, a chaotic signal tobe modulated. That is, a chaotic signal is tapped from the transmitter(in FIG. 1B, for example, the voltage at the junction of resistor 81 andcapacitor 71), applied to a communications channel, and picked up by areceiver. The switch is controllable responsively to an informationsignal, whereby the chaotic carrier signal is modulated by theinformation signal. This information signal can be detected by applyingthe chaotic signal from the channel to a receiving chaotic circuit thatsynchronizes with the chaotic signal corresponding to one of the chaoticoscillating regimes of the transmitter, but not with another anddetecting the alternations between synchronization anddesynchronization.

According to another embodiment, the invention provides a communicationsdevice with a transmitting chaotic circuit configurable responsively toan information signal. The configurations are such that the transmittingchaotic circuit produces at least three different chaotic signals, eachcharacterized by a different trajectory-versus-time characteristic. Thedevice includes a receiver with an oscillating subportion to which thedifferent chaotic signals can be applied to drive the oscillatingsubportion. The receiver has a beat detector connected to theoscillating subportion to detect a difference between a fundamentalfrequency of the oscillating subportion and a current chaotic signal.This allows the information signal to be detected by the detection ofbeats.

According to still another embodiment, the invention provides acommunications receiver with a chaotic oscillator that includes anoscillator portion and a chaotic portion. The chaotic portion has anon-linear resistance element that forms a chaotic oscillator with theoscillator portion when the chaotic portion and the oscillator portionsare coupled to pass a current signal therebetween. The oscillatorportion is signally coupled to a communications medium carrying amodulated chaotic signal. The chaotic portion is also signally coupleddirectly to the communications medium such that a voltage of thecommunications medium is directly applied to the chaotic portion througha circuit path parallel to a coupling allowing the current signal topass between the oscillator portion and the chaotic portion. Thus, boththe chaotic and oscillating portions of the receiver are driven by theincoming chaotic signal from the communications channel.

According to still another embodiment, the invention provides acommunications device with a chaotic oscillator connectable to acommunications channel. The chaotic oscillator has a tank circuit withat least two capacitors and an inductor. The first of the capacitors isconnected to an inductor and a second is selectively connectable to theinductor to combine respective capacitances of the capacitors through aswitch. In other words, the capacitors combine their capacitiesresponsively to the switch. The switch has an input for accepting aninformation signal. The information signal controls the switch so thatthe chaotic oscillator is selectively alternated between at least twooscillating regimes. The result is that a chaotic transmitter ismodulated in accordance with the information signal to generate achaotic signal which, at each instant, oscillates according to aselected one of the oscillating regimes. A receiver signally coupled tothe communications channel has a receiving chaotic oscillator portionfor each of the oscillating regimes, each portion being configured tosynchronize with a respective one of the at least two chaotic signals.By detecting which portion is in synchrony with the incoming signal, theinformation signal can be detected.

According to still another embodiment, the invention provides acommunications system with transmitting and receiving Chua circuits. Atleast one component of the transmitting Chua circuit includes at leasttwo subcomponents, at least one of which is selectively isolated fromthe transmitting Chua circuit by a switch. This is done such that acurrent oscillating regime of the transmitting Chua circuit isselectively alternated between at least two respective oscillatingregimes. The switch is switchable responsively to an information signal.The values of the subcomponents together with a configuration of theswitch are such that one of the oscillating regimes is substantially thesame as an oscillating regime of the receiving Chua circuit. The resultof the latter is that the receiving Chua circuit is synchronizable withthe transmitting Chua circuit when the current oscillating regime is thesame oscillating regime as the receiver's. A detector is connected todetect when the receiving Chua circuit is in synchrony with a chaoticsignal generated by the transmitting Chua circuit. This allows theinformation signal to be recovered from the chaotic signal (see, e.g.,FIG. 4C).

According to still another embodiment, the invention provides acommunications receiver with a chaotic oscillator that has an oscillatorportion and a chaotic portion. The chaotic portion has a non-linearresistance element that forms a chaotic oscillator with the oscillatorportion when the chaotic portion and the oscillator portions are coupledto pass a current signal therebetween. The oscillator portion issignally coupled to a communications medium carrying a modulated chaoticsignal. The chaotic portion is also signally coupled directly to thecommunications medium such that a voltage of the communications mediumis directly applied to the chaotic portion through a circuit pathparallel to a coupling allowing the current signal to pass between theoscillator portion and the chaotic portion. Thus, both the chaotic andoscillating portions of the receiver are driven by the incoming chaoticsignal from the communications channel. In this embodiment, the couplingresistance is a series of three resistors that provide the coupling fromthe voltage of the communications medium to the chaotic portion of thecircuit and provide a voltage divider network for a comparator detectorto detect voltage differences between the voltage of the communicationsmedium and the chaotic voltage generated by the receiver chaotic portionof the system (see, e.g., FIG. 4D).

According to still another embodiment, the invention provides acommunications receiver system as described in the previous paragraphexcept the receiver is divided into an oscillator portion and a chaoticportion which are separately driven through emitter followers from thevoltage of the communications medium. This allows the voltage of thecommunications medium to drive the oscillator portion and the chaoticportion of the receiver without direct feedback between the two throughthe synchronizing resistor. This prevents spontaneous chaoticoscillation in the receiver due to a feedback path from the chaoticportion of the Chua circuit to the oscillator portion of the Chuacircuit. The circuit still synchronizes since the voltage of thecommunications medium is coupled to both the oscillator portion of thereceiver and the chaotic portion of the Chua circuit. As a result, thereceiver responds only when there is a voltage on the communicationsmedium to stimulate the system (see, e.g., FIG. 4E).

According to still another embodiment, the invention provides acommunications receiver system divided into an oscillator portion and achaotic portion. The oscillator portion is driven by the voltage of thecommunications medium. The voltage of the chaotic portion is fed back tothe oscillator portion through an emitter follower and the synchronizingresistor in a phase locking type arrangement. This allows the voltage ofthe communications medium to drive the oscillator portion and thevoltage of the chaotic portion of the receiver to directly feed backthrough the synchronizing resistor to quickly synchronize thecommunications system in the presence of a voltage on the communicationschannel. This arrangement rapidly synchronizes the communicationssystem. The circuit still synchronizes since the voltage of thecommunications medium is coupled to the oscillator portion of thereceiver and the chaotic portion of the Chua circuit feeds back aportion of the receiver voltage. As a result, the receiver responds onlywhen there is a voltage on the communications medium to stimulate thesystem (see, e.g., FIG. 4F).

According to still another embodiment, the invention provides acommunications receiver system divided into an oscillator portion and achaotic portion except the nonlinear diode portion has only a Gbcomponent. This allows the oscillator portion of the circuit (e.g., FIG.41 elements 5340 and 5350) to be driven directly in accordance with theprevious embodiments discussed above. By removing the discontinuitycaused by diodes 652 and 655 (FIG. 6B), noise in the channel cannotcause the receiver to change scrolls. The circuit still synchronizessince the voltage of the communications medium is coupled to both theoscillator portion of the receiver and the chaotic portion of the Chuacircuit. As a result, the receiver responds only when there is a voltageon the communications medium to stimulate the system (see, e.g., FIG.41). The noise performance has been shown to be approximately an 8-10 dBimprovement over a receiver with the diodes in the circuit.

DETAILED DESCRIPTION First-Generation Embodiments

Referring again to FIG. 1D, discussed above, the various chaoticoscillating regimes of the Chua circuit are mapped onto an α/β parameterplane. Circuits falling in a double scroll region 60 are characterizedby oscillation about two strange attractor equilibrium points (“doublescroll”). Circuits falling in a spiral set of oscillating regimes 61exhibit oscillation about only one strange attractor equilibrium point(“single scroll”).

The oscillations of a circuit operating in a single-scroll attractormode can be seen graphically in FIG. 22A, which plots two voltages V1and V2 as a function of time for a transmitter without modulation. Thetwo voltages are measured at points 242 and 282, respectively, as shownin one embodiment of FIG. 2A. (In this example, V2 is measured at theL-C tank circuit and V1 is measured at the nonlinear resistanceelement). While the circuit operates in a non-modulated state, thevalues of V1 and V2 vary chaotically but generally swirl about anequilibrium point.

The oscillations of a circuit operating in a double-scroll attractormode can be seen graphically in FIG. 20A, which plots the same twovoltages V1 and V2 as a function of time for a circuit that is notmodulated. While the circuit operates in a non-modulated state, thevalues of V1 and V2 vary chaotically but generally swirl about twodistinct equilibrium points.

Varying C₁ causes the α/β combination to shift as indicated by arrow 130in FIG. 1D. Varying C₂ causes the α/β combination to shift as indicatedby arrow 140. Varying R₂ or L causes the α/β combination to shift asindicated by arrow 150. As can be seen from the diagram (FIG. 1D), thecapacitance C₂ of capacitor 20 can be varied over a wide range whilestill maintaining operation of the circuit in the double scrolloscillating regime 110. A much smaller range of values of capacitance ofcapacitor 15 (C₁) coincides with operation in the double scroll regime60. The wide range of capacitances for C₂ (capacitor 20) that coincidewith operation in the double scroll region 60 is exploited in a firstembodiment of the invention discussed immediately below.

Referring now also to FIG. 2A, in a first-generation embodiment of theinvention, a transmitter 200 includes a modified-Chua circuit. Thetransmitter 200 generates a modulated chaotic signal responsively to aninformation signal 236. The transmitter 200 has a primary 220 andauxiliary 237 capacitor. The auxiliary capacitor 237 is selectivelyswitched into the circuit to add selectively to the C₂ capacitance ofthe embedded Chua circuit. By switching the auxiliary capacitor 237 offand on, the transmitter oscillates according to a base oscillatingregime and an alternate oscillating regime, respectively. By controllingoptoisolator 235 responsively to an information signal 236, analternating pattern of chaotic oscillations is generated which can becharacterized as a modulation of the base chaotic oscillation. Thismodulated chaotic pattern can be transmitted to a receiver bytransmitting a voltage V₁ tapped from point 242.

Referring now also to FIG. 2B, the modulated chaotic signal is detectedby a receiver 201 containing a modified-Chua circuit whose componentproperties are chosen to insure that the receiving circuit 201 willexhibit the same oscillating behavior as the base configuration(auxiliary capacitor 237 switched off) of the Chua circuit of thetransmitter 200. The need to match oscillating behaviors is to allow thereceiving circuit 201 to synchronize with the received signal 291. Oneway to match the oscillating behaviors of the transmitting and receivingcircuits 200 and 201 is to match the values of the components thatdetermine the oscillating behavior. The resulting transmitted voltageV₁, output from the transmitter 200, is applied as input signal 291 tothe receiver 201. Note that the component values need not be matchedperfectly. It has been found that the receiver's α and β can differ byas much as approximately 5 percent from the transmitter's withoutsubstantially affecting the ability of the circuits to synchronize.

The use of an optoelectronic switch 235 avoids any need for outputbiasing. Also, an optoelectronic switch 235 also adds no significantcapacitance to the circuit. A low output biasing and low capacitance ofthe switching element make it easier to match the component values ofthe receiving and transmitting circuits 201 and 200 to insuresynchronization. Alternatively, a reed switch or a field effecttransistor (FET) can be used to isolate the auxiliary capacitor 237 fromthe main circuit. To provide a lower output biasing requirement,multiple FETs can be employed as a single switch. In the embodiment ofFIGS. 2A and 2B, the modulated chaotic signal is produced by varying thecapacitance C₂ of the tank circuit 231 as described above. That is, inthe transmitter 200, the auxiliary capacitor 237 is isolated from themain circuit by an optoelectronic switch 235, which effectively changesthe capacitance of capacitor 220 in FIG. 2A.

To modulate the transmitter 200, the capacitance C₂ of the tank circuit231 is modulated by intermittently combining the capacitance ofauxiliary capacitor 237 with that of capacitor 220. This capacitancecorresponds to the capacitance of capacitor 20 in the unmodified Chuacircuit 101; that is, to C₂. By intermittently altering this capacitanceresponsively to the input signal 236, the Chua circuit of thetransmitter 200 alternates between two different oscillating patterns.

The voltage signal V₁(t) can be transmitted by any means desired. Forexample, the output chaotic signal V₁(t) can be used to modulate anoptical carrier, laser carrier, radio carrier, applied directly to ametallic (wire) interface, applied to a speaker and transmitted as soundwaves, or transmitted using any other mechanism. The received signal canalso be applied through an automatic gain control circuit (not shown inthis embodiment) for signal conditioning.

In the embodiment of FIG. 2B, the received signal can be applied througha voltage follower 251, if desired for high input impedance, through aresistor 280, and finally to a bridge point 281 of L-C tank circuit 261.Tank circuit 261 has an inductor 248 and a capacitor 260 and can includea resistor in addition to its inherent resistance. The L-C tank circuitis connected to the chaotic portion of the embedded Chua oscillator ofthe receiver circuit 201 by a voltage follower 245. Current from the L-Ctank circuit 261 is applied through the voltage follower 245 and aresistor 265 whose resistance matches that of the transmitting circuitresistor 225. In other words, the resistance R of the Chua circuitsmatch.

By matching the resistance of 280 to that of resistor 225, tank circuit261 is driven or pumped by the incoming signal exactly as tank circuit231 is pumped by chaotic portion 222. When the C₂ values of thetransmitter tank circuit 231 and the receiving tank circuit 261 areidentical (that is, when auxiliary capacitor 237 is isolated from thetank circuit by opto-isolator 235), a time-varying voltage at 281synchronizes and subsequently tracks that of the incoming signal 291.This synchronization occurs because the incoming signal matches that atcorresponding point 282 of the transmitting circuit, so the environmentsof tank circuit 261 and tank circuit 231 are the same.

Also, the voltage at point 281 is applied through a resistor 265 that isalso matched to resistor 225 so the environment of the chaotic part 262of the receiving circuit 201 is also the same as the environment of thechaotic part 222 of the transmitting circuit 200. Thus, when thetransmitter is oscillating about the base strange attractor equilibriumpoints (base referring to the situation when the auxiliary capacitor 237is isolated from the transmitter 200 so all the circuit elements of thetransmitting 200 and receiving 201 circuits match), tank circuit 261quickly goes into an oscillating pattern that is in synchrony with thatof the transmitter's tank circuit 231. When the auxiliary capacitor 237is switched on by closing the optoisolator 235, the transmitter circuit200 oscillates in a pattern that is no longer matched to that of thereceiver and the receiver 201 can no longer track the signal perfectly.That is, the transmitting circuit 200 and the receiving circuit 201 nolonger synchronize. Note, the value of the combined capacitance C₂ canbe varied over the range 1 μF to 0.015 μF, a dynamic range of 66:1.

As discussed above, in one embodiment, receiving circuit 201 and thebase configuration of the transmitting circuit 200 can be preciselymatched, in terms of their oscillating behavior, to insure that receiver201 will alternately synchronize and go out of synchronizationresponsively to the transmitter 200. That is, according to thisembodiment, the transmitter's and the receiver's behaviors must besubstantially matched for the transmitting and receiving circuits 200and 201 to form an effective communications device. Since β (See FIG.1D) varies as the square of resistance, precise resistors should be usedin the transmitter 200 and the receiver 201. A combination of a 1580 ohmfixed resistor 280 and a 200 ohm, 25-turn pot 265 can be used forresistors 280 and 265 in receiver 201. This allows tuning of thereceiver resistors to obtain a precise match to those in thetransmitter. Note, if component values of the receiving circuit arechosen to match the values of α and β of the transmitting circuit butdepart more substantially from those of the receiving circuit, thereceiving circuit can still be driven into synchronization, but theresponse will not be as strong.

Note that the receiver circuit with the synchronizing resistor added canbe configured with components that permit the receiver to produce adetectable pattern at the detector output of a chaotic signal producedby a Chua receiver whose component values do not match the transmittingcircuit's precisely. A fully functional communication system can be madesince the receiver can produce consistent output beat frequencies fromthe detector while the receiver circuit tries to follow the inputsignal. The receiver therefore detects signals that are not synchronizedwith the receiver chaotic parameters through a received process thatshifts the amplitude and phase of the incoming strange attractor. Thismode of operation produces a response over a wide range of α/β values aswell as frequency ranges for a single specified α/β combination. One canbuild a vocabulary using this technique even when the chaotic signalattractor frequency is varied and α/β are different. The synchronizingresistor produces this beat frequency effect at the output of thedetector.

A comparator 270 detects the alternating pattern of synchronizations anddesynchronizations of the receiving circuit 201. The received signal at286 is compared by comparator 270 with the voltage generated at 287.These two voltages are at corresponding locations 242 and 287 in thetransmitting 200 and receiving 201 circuits. When the transmitting 200and receiving 201 circuits are in synchrony, that is when the voltagesfollow the same time-dependent trajectory pattern, the comparator output290 is zero. When the transmitting 200 and receiving 201 circuits arenot in synchrony, that is when these voltages do not follow the sametime-dependent trajectory pattern, the comparator output 290 isnon-zero. A waveform is generated characterized by pulses representingthe points where the two waveforms diverge by some voltage differencedetermined by the sensitivity of comparator 270. The sensitivity(voltage difference trigger) can be set by replacing the fixedcomparator 270 with a comparator having hysterisis. Of course there canbe a DC component, but that is ignored or filtered out of the comparatoroutput 290.

The following is what occurs in the transmitter as the transmittingcircuit 200 is modulated. When the auxiliary capacitor 237 is switchedout of the circuit during a first time interval, the transmittingcircuit oscillates around the base (strange) attractor at a particularattractor frequency. The circuit, of course, oscillates not at aspecific frequency, but about a set of attractors. That is, themodulated signal is chaotically “smeared” over a range of frequenciesaround the current strange attractor equilibrium point or points.

In the embodiment of FIGS. 2A and 2B, the equilibrium point(s)correspond to the base value of C₂, which corresponds to the capacitanceof capacitor 220 alone. Note that for some of the operating regimesillustrated in FIG. 1D, the equilibrium point(s) can be dependent on theinitial state also. When auxiliary capacitor 237 is switched on by theopto-isolator 235, the transmitting circuit 200 begins oscillating in adifferent pattern corresponding to a then-current initial state and thenew value of C₂ equal to the capacitance of capacitor 220 and auxiliarycapacitor 237.

When a modulation pattern of successive actuations/deactuations isapplied through opto-isolator 235, the transmitting circuit 200 switchesbetween the first (base) signal and the second signal in accordance withthe modulation pattern. In the receiver 201, the transmitted V₁(t)signal is applied at 281 and through resistor 280. The receiver's L-Ctank circuit 261 sees the received V₁(t) applied through resistor 280 sothat when the base-signal is transmitted, the V₁(t) signal applied at286 causes the voltage at 281, V₂′(t), to follow the same time-dependenttrajectory as V₂(t) of the transmitting circuit 200. Given that V₂′(t)in the receiving circuit is substantially identical to V₂(t), the V₁′(t)in the receiving circuit follows the same time-dependent trajectory asV₁(t) of the transmitting circuit 200. Thus, the two circuits aresynchronized when the transmitting circuit is oscillating about the basestrange attractor(s). When V₁(t) received corresponds to the basesignal, V₁(t) and V₁′(t) are substantially identical and an output 290of comparator 270, to whose inputs V₁(t) and V₁′(t) are respectivelyapplied, has a nominally zero amplitude. In a practical system, theoutput of the comparator can have some hysteresis so that the differencemust exceed some nominal level to generate a nonzero output.

When optoisolator 235 switches in the auxiliary capacitor 237,transmitting circuit 200 oscillates about the second strangeattractor(s). When that happens, the receiving circuit 201 can no longersynchronize in response to the V₁(t) applied at 281. This is because C₂′(the capacitance of capacitor 260) no longer matches C₂ (the combinedcapacitance of capacitor 230 and auxiliary capacitor 237). As a result,the pattern of oscillation of V₁′(t) is no longer synchronized withV₁(t) transmitted and the output 290 of comparator 270 is no longerzero. The changes in output 290 between the nominal zero and nominaloscillating states can be registered through some convenient system,such as a power meter or frequency counter on the output 290, togenerate a signal carrying the data in the information signal 236.

The receiving circuit 201 is very sensitive to the chaotic signal ofV₁(t). That is, the incoming signal can be a very low peak power signal(power within a narrow frequency band about a peak) while still causethe receiving circuit to synchronize with the transmitting circuit.Thus, receiving circuit 201 is able to follow the distinct chaotictime-dependent pattern of the received signal (when the base signal isbeing transmitted) and only by virtue of its inherent tendency to follow(be driven by) it, is the receiver 201 able to pick up the signal out ofthe noise. It is very much a resonance phenomenon, even though it is notperiodic in the sense of strict linear oscillatory phenomena. This kindof resonance effect allows the power to be so spread over a range offrequencies that the transmitted signal appears to be “buried” in thenoise of the channel. The signal can be detected by means of a receivingcircuit whose parameters are closely matched with the transmittingcircuit or by applying a large FFT engine (1024 point or larger) to thesampled data. The values of the components of the transmitting circuitmust be known in order to pick up the data signal masked in the chaoticsignal applied to the channel. The sensitivity of the variousembodiments described below is so great that an effective communicationssignal can be characterized by a zero dB signal-to-noise ratio.

Note that a higher signal-to-noise ratio will allow synchronization tobe achieved with less precise component matching. To achieve reliabletransmission with the lowest signal-to-noise ratio, the precision ofcomponent-value-matching should be as high as practical. To achieveprecise resistance matching, as mentioned, fixed resistors can be usedwith 25-turn fine-tuning resistors, either in parallel or series foreach resistor 280 and 265.

As discussed above, a large number of different operating regimes areavailable by modifying C₂. However, the receiver 201 may not be capableof distinguishing among these because of the inability of the receivingChua circuit to track a received signal 291 other than the one generatedby the base configuration of the transmitting circuit.

Referring now to FIGS. 3A and 3B, a transmitter 300 has a bank ofauxiliary capacitors 237 a-237 n that can be selectively switched intothe tank circuit portion of the transmitter 200′ by respectiveopto-isolators 235 a-235 n. Opto-isolators 235 a-235 n are controlled bya controller 201 responsively to the input signal 236. In all otherrespects, transmitter 200′ is identical to transmitter 200 of FIG. 2A.

To produce a modulated signal, controller 201 is programmed to receive adata word at 236 and control opto-isolators 235 a-235 n to switch one ormore of them into the circuit to produce a particular chaotic pattern.For example, if opto-isolators 235 a and 235 b are actuated to switch inauxiliary capacitors 237 a and 237 b, the capacitances of capacitors220, 237 a, and 237 b are combined to produce a corresponding chaoticsignal. This unique pattern forms one word or symbol of a vocabulary ofpossible ones that can be generated by actuating opto-isolatorsaccording to the various possible combinations.

Receiver 301 includes a bank of receivers 201 a-201 n, each the same asshown in FIG. 3B but with different component values chosen to alloweach to resonate with a corresponding configuration of the transmitter200′. That is, the component values for each receiver 201 a-201 n, arechosen such that each will synchronize with one word of the vocabularyof chaotic signals generated by the transmitter 200′. The outputs ofeach receiver 201 a-201 n each correspond to the output 290 of thecomparator of FIG. 2B. These are individually applied to a detector 271that determines which of the outputs 290 a-290 n is in synchrony withthe transmitter 200′ at a given point in time. Detector 271 determinesthis by indicating which output 290 a-290 n is nominally equal to zero.

Output 290′ can be a stream of digital words each corresponding to theword transmitted. Thus, at any given instant, signal 290′ can indicatethe current, or most recently received, word being applied in thereceived signal to input 291′. The number of bits that can betransmitted with each cycle of the opto-isolators 235 a-235 n is equalto the log₂ (log base-2) of the number of different capacitor values C₂forming the vocabulary. In this embodiment, the number of receivers 201a-201 n matches the size of the vocabulary, that is, the number ofsymbols or words generated by the transmitter 200′.

Referring to FIGS. 4A and 4B, an alternative embodiment of the inventionalso produces a vocabulary of signal-words. In this embodiment, thevalues of the various components are chosen so that all “words” of thevocabulary are produced by configurations that maintain the same α-β,combinations. That is, the values of C₁, C₂, R, and L of thetransmitting circuit 400 are varied to produce a variety of selectablechaotic oscillation patterns, each of which coincides with the same α/βcombinations. A bank of parallel inductors 330 can be provided that canbe switched in by respective opto-isolators to add to the inductance ofthe fixed inductor 330′. To this end, a bank of parallel capacitors 320can be provided that can be switched in by respective opto-isolators toadd to the capacitance of the fixed capacitor 320′. A bank of parallelcapacitors 315 can be provided that can be switched in by respectiveopto-isolators to add to the capacitance of the fixed capacitor 315′.Finally, a bank of parallel resistors 325 can be provided that can beswitched in by respective opto-isolators to change the resistance offixed resistor 325′.

In the embodiment of FIGS. 4A and 4B, instead of modulating the behaviorof the transmitting circuit by varying only one component, the values ofcombinations of components determining C₁, C₂, R, and L are varied insuch a way as to maintain α=C₂/C₁ and β=R²C₂/L constant under thecontrol of a controller 305. That is, values of the auxiliarycapacitors, resistors, and inductors in banks 315, 320, 325, and 330 arechosen so that when predefined combinations of the correspondingopto-isolators are actuated, the oscillating frequency changes (Seeequation (1), below), but α and β remain the same.

In a first embodiment, the inductance L and the capacitances C₁ and C₂,only, are varied, maintaining R constant. That is, only theopto-isolators in banks 315, 320, and 330 are switched to modulate thecircuit. The fundamental frequency of the tank circuit, given byF=1/[2π(LC ₂)^(1/2)]  (1)varies even though the oscillating regime remains constant. In thisembodiment, there is no need to vary R to maintain a constant α and β.Also, the non-linear resistor 310 components do not have to be varied.In one embodiment of the transmitter 400, one combination of values ofC₁, C₂, and L (corresponding to one state of the opto-isolator switches)can be identical to the combination of values corresponding to areceiving circuit 301. This combination of values causes thetransmitting circuit to oscillate at a base frequency corresponding tothe frequency of the receiver. In a first alternative embodiment, thevalues of C₁, C₂, and L are varied, keeping R constant.

The receiver 401, shown in FIG. 4B, consists of a tank circuit 361 and afast Fourier transform FFT computer 379. The component values of tankcircuit 361 are chosen to match the α/β combinations of the transmitter400. The value of resistor 380, in the first embodiment, can be chosento match that of the fixed R value of the transmitter. (As mentionedabove, R is fixed in the first embodiment. The embodiment depicted inFIG. 4 a is capable of variable R values, but a single fixed value isused in the first embodiment. If a fixed R value is to be used, the bankof resistors 325 could be omitted leaving only a single resistor 325′.)

The output 390 of the transmitter 400 is applied (through sometransmission medium) to an input 391 of the receiver. The FFT block 379determines the word transmitted by the transmitter by detecting thedifference between the fundamental frequency, given by equation (1), ofthe instant configuration of the transmitter 400, and the fixedconfiguration of the tank circuit 361. Fast Fourier transform (FFT)block 379 “sees” this frequency difference because tank circuit 361attempts to follow the trajectory of the incoming signal applied throughresistance 380. Since, however, the incoming signal is at a differentfrequency, tank circuit 361 is not able to maintain this trajectory andit “falls off the rails.” The frequency with which the tank circuitalternately follows and uncouples from the incoming signal applied at391 is equal to the difference between the fundamental attractorfrequency of the transmitter 400 and that of the tank circuit 361. Apeak at this difference frequency is indicated by the output 378 of theFFT block. Thus, output 378 indicates each word generated by thetransmitter 400.

In an alternative embodiment, the value of R (the resistance determinedby the configuration of resistor bank 325) of the transmitter 400 isvaried also. In this case, the value of resistor 380 of the receiver 301can be any of the values used for R in the transmitter 400 or anothervalue.

In the receiving circuit 201 of FIGS. 2A and 2B, the Chua circuitoscillates in synchrony with that of the transmitter 200 when the basesignal is received. The resulting output 290 from the comparator 270, asdiscussed, is a flat (nominally zero) output. However, when thetransmitter 200 is placed in a configuration such that it a signal otherthan the base signal, the comparator output 290 of the receiver 201becomes substantially non-zero and, also, chaotic due to the lack ofsynchronization. This would also be the case if the transmitter 400 wereconnected to the receiver 201 of the prior embodiment. Any signal otherthan ones that drive the receiver into synchronization will produce anon-zero chaotic signal at the output 290. In other words, if thisoutput 290 were viewed on an oscilloscope, the trace would appearchaotic and it would be difficult if not impossible to tell which “word”of the vocabulary of strange attractors was generating the transmittedsignal. This is because V₁′ can no longer synchronize with the V₁transmitted.

In the combination of the transmitter 400 and the receiver 301 of theFIGS. 4A/4B embodiment, however, the transmitter drives the tank circuit361 for part of the tank circuit's fundamental cycle but subsequentlyskips ahead or lags behind (unless, of course, the fundamental frequencyof the transmitter is the same as that of the receiver) the drivinginput voltage applied at 391. This happens at the frequency difference(a beat frequency; the difference between the fundamental frequency ofthe transmitter (e.g., (1)) and the fundamental frequency of thereceiver) and thus forms a detector. The reason the receiver is able tofollow the transmitter in this way is that the α and β of thetransmitter and receiver are matched and the natural frequency of thetank circuit is an integer multiple of the frequency of the transmittingcircuit given by equation (1).

In addition to using a FFT calculator, alternative ways of detecting thebeat frequency include an amplitude detector connected across the sameterminals as the FFT calculator and which outputs to a counter. Thecounter can count the number of beats to determine the frequency of thetransmitter. Various other alternatives would occur to a practitionerskilled in the art of signal conditioning.

Referring now to FIGS. 4A and 4C, an alternative receiver design employsa synchronizing resistor 385 and comparator input resistors 386 and 387.In this embodiment, the received signal, corresponding to a signal otherthan the base strange attractor, is applied to the chaotic portion 362through the synchronizing resistor 385. In the embodiment of FIG. 4C, ithas been found, through experiment, that the synchronizing resistor 385can be adjusted to optimize the ability of the receiver circuit 402 tofollow the trajectory of the transmitter signal at portions thereof sothat the zero levels of the output 390 are clean and clear. Theapplication of the received signal to the chaotic portion 362 of thereceiver 402 through synchronizing resistor 385 forces the chaoticportion into an oscillation pattern that forms clear and distinct beatswith the original transmitted signals that are output by the comparator370. Because the beats (amplitude differences) are distinct and clear, acounting circuit 392 can be used to indicate the detected word. As inthe embodiment of FIG. 4B, the beats indicate the frequency of thetransmitting circuit 400 oscillations. It appears the reason thesynchronizing resistor 385 allows clean beats to be formed is that thereceiver is forced by the application of V₁(t) to the chaotic portion tofollow certain portions (i.e., zero crossings) of the chaotic signalfrom the transmitter 400. During these brief intervals, the comparatoroutput 390 goes to nominal zero output.

Counting circuit 392 can be any of various circuits for determining thenumber of such zero-intervals per unit time. For example, the countingcircuit can reset a timer at the start of a nominal zero (zero within atolerance) and output a pulse to a counter if the zero is maintained forsome threshold interval. The counter can be reset through thetransmission of a base chaotic signal for which the comparator outputremains zero for a second threshold interval. The counter canautomatically reset at predetermined intervals maintained in synchronyby the transmission of the base chaotic signal.

Note that the use of a photo field-effect transistor (FET) optocouplerfor the opto-isolator in the various transmitter embodiments isrecommended because of the following properties. First, the example usedfor evaluating the circuits tested has a gallium-arsenide infraredemitting diode coupled to a symmetrical bilateral silicon photodetector.The detector is electrically isolated from the input signal and performsas an ideal FET. Distortion free control of low level alternatingcurrent (AC) and direct current (DC) signal is enabled. The primaryfeature that relates to modulating chaos is the low output impedance ofthe FET when active (i.e., ≦100 ohms). Also, the device exhibits highimpedance (≧300 Mega-ohms) when switched off.

The output FET is controlled via a channel voltage that does not requireexternal biasing of the device so that it can operate as an idealswitch. This is compared to conventional FETs that require externalbiasing. In addition, the device is characterized by a shunt capacitanceof ≦15 picofarads. The circuits allow switching speeds of up to 2Megahertz. For slower speeds a solid state relay switch can be used.Finally, a normal FET has some capacitance between the drain and thegate. The small current differential permitted by this can prevent thecircuit from operating chaotically or it can make it difficult to matchthe chaotic pattern of the base signal in the receiving and transmittingcircuits.

The embodiment of FIG. 4D, receiver 403, has the characteristics of thatof FIG. 4C except the synchronizing resistor is formed of the sum ofresistors 385, 387, and 388. As in the embodiment of FIG. 4C, theembodiment of FIG. 4D also uses a simple counter circuit for determininga beat frequency. Also, the combination of resistors 385, 387, and 388provide a synchronizing element to lock the chaotic portion of thereceiver to the incoming signal, as in the embodiment of FIG. 4C. Inthis embodiment, however, a voltage follower 351 isolates the incomingcommunication signal from the receiver-generated chaotic signal.

It has been determined that a resonance voltage difference is achievedwhen the resistance of the synchronizing resistor is approximately 5times the resistance of resistor 380. This configuration maximizes thevoltage difference between points 286 and 287 of FIG. 4D and FIG. 2B,respectively.

The embodiment of FIG. 4E (receiver 404) adds an emitter follower 353 toisolate the oscillator portion 361 from point 287. In other words, avoltage follower blocks feedback from point 287 to the oscillatorportion. It provides feed-forward coupling of the incoming signal 391.This causes synchronization to be forced through two parallel paths. Onepath is through the emitter follower 351 and synchronization resistors385, 388, and 387 to the chaotic portion of the Chua circuit. The otherpath is through the oscillator portion 361 and the emitter follower 345and resistor 365.

In the FIG. 4F embodiment, receiver 405 reverses the feedback path usingan emitter follower 363 to provide a phase lock loop type operation inwhich the voltage V₁′ is fed back into the oscillator portion of theChua circuit to achieve synchronization. In this embodiment, voltagefollower 363 isolates the receiver-generated signal from the incomingsignal and allows the receiver generated signal to feed back into theoscillator portion of the Chua circuit to cause faster synchronization.

Referring now to FIG. 5, a general chaotic communications system inwhich the synchronization concept identified above is applied includes atransmitter 580 and a receiver 590. Transmitter 580 includes a firstsubsystem 500 signally coupled to a second subsystem 505. Subsystems 500and 505 can include common elements, but are not coextensive. Firstsubsystem 500 applies a signal S₁ to second subsystem 505 and secondsubsystem 505 applies a signal S₂ to first subsystem 500. Firstsubsystem 500 drives second subsystem 505 with signal S₁ and secondsubsystem 505 drives first subsystem 500 with signal S2.

Either first subsystem 500 or second subsystem 505 can be driven by adriving signal or by some current source 515 or simply by establishingan initial state if both systems are lossless as in a purely numericalsystem generated by a computer. Current source 515 could form a portionof either subsystem 500 and 505 as in a Chua diode described in detailin various prior art publications, for example, Chua's Circuit: AParadigm for Chaos, Ed. Rabinder N Madan, (see pp. 13-24), WorldScientific Publishing, NJ, USA, 1993; the entirety of which book isincorporated herein by reference. Alternatively, in a physical systemsuch as an electrical circuit, current source 515 could be derived froma driving circuit such as described in U.S. Pat. No. 5,473,694 (element115 in FIGS. 1 and 7) the entirety of which patent is incorporatedherein by reference. Again in a numerical model, current source 515would be absent since the transmitter 580 could be considered lossless.

The transmitter is perturbed by some means to modulate the signal S₂ toconvey information. This can be done by injecting current into eithersignal S1 or S2 or some portion of either or both of the first andsecond subsystems 500 and 505 or by modifying parameters of either orboth of the first and second subsystems. This information signalinjection process is represented by the application of informationsignal S_(i) to either of the first and second subsystems 500 and 505.Note, however, that the information signal can be coextensive with adriving current source 515 such as described in U.S. Pat. No. 5,473,694and that it can be applied as a current addition to either or both ofsignals S₁ or S₂ or by application to either of subsystems 500 or 505.

In accordance with one aspect of the invention, signal S₂ is applied tothe receiver 590. In the prior art arrangements, the signal S₂ would beapplied only to a third subsystem 530 whose configuration closelymatches first subsystem 500. Transmitted signal S₂ is also appliedthrough a synchronizing filter 550 to the fourth subsystem 535 whosecharacteristics closely match those of second subsystem 505. To extractthe received signal, the incoming signal S₂ is processed together withthe signal S₄ generated in the response, for example by subtracting themin a comparator with hysteresis 540. Alternative methods of processingthe received signal and the signal S₄ generated by the receiver can beemployed, for example, the phase comparison technique described in U.S.Pat. No. 5,473,694.

Note that in all of the above embodiments, it may be desirableperiodically to send a registration signal, for example, a contiguousseries of base chaotic signals, to insure that any clocks on thereceiving and transmitting ends are aligned. Such registration might beneeded, for example, to demarcate the time blocks applied to the FFTcalculator so that the correct time series is sampled. This might not benecessary, depending on the size of the FFT block relative to the symbolduration.

Note that the invention can be implemented using a computer rather thandiscrete components since the Chua circuit is readily susceptible todigital simulation. In this case, all the above-described processescould be implemented digitally, with appropriate use of A/D and D/Aconversion at either end of the communications channel. Thecommunications systems described also form the basis of computeralgorithms that can be implemented in a processing architecture toimplement a chaotic communications system. References to discretecomponents and time-varying parameters should be interpreted toencompass their analogues in the digital signal processing domain.Similarly, references herein to “circuits” should be understood toinclude both analog and digital implementations of the circuits.

II. SECOND-GENERATION EMBODIMENTS AND TECHNIQUES

As described above, various first-generation systems use modulatedchaotic circuits to transmit information in a low-power, high-noiseenvironment with simple circuit elements. Modulated chaoticcommunication systems such as those described above provide benefitssimilar to spread-spectrum technology (e.g., covert communicationapplications and noise immunity) using a simpler, cheaper circuit designand improved signal-to-noise ratios.

The present inventors have discovered that the modulation techniquesused in first-generation systems may be subject to certain bandwidthlimitations. In particular, it has been determined that the modulationbandwidth using the aforementioned techniques may be generally limitedto 10 to 15% of the tank circuit frequency in the transmitting circuit.This limitation is believed to be due to the fact that changing lumpparameters (reactive components) in the transmitter requires a certainamount of settling time in the transmitter before the receiver cansynchronize with the changed transmitter parameters, although thistheory is not critical to understanding or practicing the presentinvention. The reactive components' impedance change is believed tocause signal spikes that ring the receiver at the discontinuity pointsthat cause receiver noise spikes.

Modulation bandwidth is an important consideration in applying theprinciples of the invention commercially. In cellular telephone systemsand cable television transmission systems, for example, the ability totransmit larger quantities of information over a channel of a givenbandwidth translates into lower costs. Consequently, it is desirable toincrease the bandwidth when modulating chaotically oscillatingtransmitters.

In order to overcome the aforementioned limitations, the presentinventors have discovered that by modulating certain characteristics(including nonreactive components) of the transmitting circuit, themodulation bandwidth can be increased by approximately 200%. In someembodiments, this effect occurs as a result of changing one or morenon-reactive resistive values in the transmitter circuit, which allowsthe transmitter to smoothly transition between strange attractors, whichcauses the receiver to go into and out of synchronization almostinstantly without generating noise. Various transmitter circuitsaccording to the second-generation embodiments can be used withfirst-generation receivers described above. Other features, improvementsand advantages of the second-generation system will become apparentthrough the following description and accompanying figures.

A. General Principles

Turning first to FIG. 19A, a second-generation system employing variousinventive principles is shown. The system includes a chaotic transmittercircuit 1901 coupled to a communications channel 1902 and a chaoticreceiver circuit 1903. Like the chaotic transmitters described abovewith respect to the first-generation system, transmitter 1901 includes atank circuit characterized by inductor L and capacitor C2, a resistorR₀, and a chaotic portion including a second capacitor C1 and anonlinear element 1904.

As in many of the first-generation systems, nonlinear element 1904includes a negative resistance element having a current-voltage responsecharacteristic such as that illustrated by line 1910 in FIG. 19B. Incontrast to the first-generation systems, however, the current-voltagecharacteristics of nonlinear element 1904 are modulated in accordancewith an information signal s(t). This modulation behavior is denoted byan arrow through the nonlinear element as shown in FIG. 19A. In oneembodiment, nonlinear element 1904 comprises a negative resistancediode.

As with the first-generation systems, communications channel 1902 cancomprise any of various transmission media such as wire, light(including fiber optic), radio frequency (all bands), or sound, forexample. Chaotic receiver circuit 1903 can comprise any of the variousreceivers described above with respect to the first-generation systems,or may include various enhancements described below.

FIG. 19B shows the current/voltage (I/V) characteristics of a nonlinearelement such as those used in the first-generation system. A directcurrent load line 1911 is superimposed over a piecewise linearcurrent-voltage characteristic line 1910, representing the quantity(−1/R), where R is a resistance coupling the tank circuit to the chaoticportion of the circuit as shown in FIG. 19A.

FIG. 19C shows two single scroll attractors 1920 and 1930 orbitingaround equilibrium points where nonlinear element characteristic curve1910 intersects load line 1911. When operating in a single-scrollattractor mode, the current-voltage point oscillates about eitherequilibrium (“attractor”) point 1920 or 1930, depending on circuitparameters. When operating in a double-scroll attractor mode, thecircuit transitions between upper attractor 1920 and lower attractor1930 of the curve when the strange attractor voltage touches or crossesthe breakpoint voltage −Bp or +Bp (i.e., a discontinuity point on the IVcharacteristic curve defined by the diode breakdown voltage +/−Bp andany biasing voltage applied to the diodes). These principles weredescribed generally in connection with the first-generation systemsabove.

According to various second-generation embodiments, the equilibriumpoints are shifted in accordance with an information signal by changingone or more resistance values of nonlinear element 1904. As shown inFIG. 19D, modulating these resistance values has the effect of shiftingthe strange attractor points to different equilibrium positions alongthe load line. In other words, an information signal is used to shiftthe equilibrium points, changing the strange attractor frequency andamplitude. Because this shift can be accomplished by changing anonreactive resistance value, the shift occurs quickly, instantlyshifting the transmitter's operating regime between different strangeattractors and allowing a receiver to synchronize very quickly (e.g., inless than a quarter cycle of the tank circuit's fundamental frequency).

FIG. 19E shows the strange attractors moving with the equilibrium point,wherein the impedance of the load is changing and a time varying signalis traced out on the voltage across the nonlinear diode input. As shownin FIG. 19E, three or more different symbols can be generated, eachcorresponding to a single equilibrium point on the characteristic curve.

FIG. 19F shows one technique for changing the nonlinear diodecurrent-voltage (IV) characteristic curve using an ideal switch (SW1 andSW2) and a resistor in series with the ideal switch (R5) placed inparallel with one of the nonlinear diode resistors. Further details ofthis circuit are provided below. In general, FIG. 19F shows a so-called“Caltech” diode that is modulated such that the slope Gb of thenonlinear characteristic curve 1910 in FIG. 9D is changed. This causesthe strange attractor to change frequency and equilibrium points asshown in FIGS. 19D and 19E. Gb changes as follows:Gb=R ₃ −R ₁ /R ₃ ×R ₁When R5 is placed in parallel with R3 the new Gb value is:${Gb} = \frac{\frac{R_{3} \times R_{5}}{R_{3} + R_{5}} - R_{1}}{\frac{R_{3} \times R_{5}}{R_{3} + R_{5}} \times R_{1}}$This yields a new slope for Gb and this new slope crosses load line 1911in FIG. 19D at points 2 or 3.

FIG. 19G shows the result of modulating the voltage across the nonlinearelement. Note that both the amplitude of V1 and the frequency arechanged. At the receiver, the frequency and/or amplitude changes can beused to demodulate the signal as shown in various first-generationembodiments.

In general, chaotic receivers described above with reference to thefirst-generation systems can be used to detect and recover informationtransmitted with second-generation transmitters. As one example, thereceiver shown in FIG. 4C can synchronize to a transmitter and, when thetransmitter's strange attractor is changed in accordance with theprinciples described above, the receiver will fall out ofsynchronization. These changes can be detected by a counting circuit orany of various other methods described in the first-generation systemsin order to recover information.

In general, changing the current-voltage characteristics of a circuitelement in a chaotic transmitter by changing a nonreactive circuit valueto transmit information will be referred to as “chaotic trajectory shiftmodulation” and the resulting signals will be referred to as “chaotictrajectory shifted” or “chaotic trajectory modulated” signals. In thespecific case where a chaotic signal is transitioned between two or morediscrete signal profiles by such techniques, the modulation will bereferred to as “chaotic trajectory shift keying modulation” and theresulting signals will be referred to as “chaotic trajectory shiftkeyed” signals.

B. Modulating a Nonlinear Element of the Transmitter

Techniques for modulating characteristics of a nonlinear element in achaotic transmitter will now be described according to varioussecond-generation embodiments of the invention.

FIG. 6A shows a chaotic oscillator including a tank circuit 231,capacitor 215, resistor 225, and a so-called “Kennedy” diode 600 thatcan be used as the nonlinear element of a Chua transmitter shown in FIG.2A. This diode, which is known in the prior art (see, e.g., Chua'sCircuit: A Paradigm for Chaos, Rabinder N. Madan, pp. 86-87 (1993)),includes op amp 602 coupled to resistors 601, 603, and 604, and op amp605 coupled to resistors 606, 607, and 608. References herein to“Kennedy diode” should be understood to refer to subcircuit 600 of FIG.6A and its many obvious variations. Tank circuit 231, resistor 225, andcapacitor 215 are the same as or similar to those shown in thetransmitter circuit of FIG. 2A.

In general, Kennedy diode 600 of FIG. 6A is characterized by a piecewiselinear current/voltage function across its terminals as illustrated inFIG. 8 by segments 860 a, 860 b, and 860 c (the entire line will bereferred to as element 860). The slope of segments 860 a and 860 c,referred to as Gb, is defined by various resistive values within thenonlinear element as described below. Similarly, the slope of segment860 b, referred to as Ga, is defined by resistive values within thenonlinear element. The negative breakpoint −Bp and the positivebreakpoint Bp, which define where the slopes changes occur, are alsodetermined by resistive values in nonlinear element 600. It will beappreciated that the piecewise linear characteristics illustrated inFIG. 8 represent only one of several different possible nonlinearcircuits.

Load line 850 represents a current-voltage characteristic of a resistiveelement coupling the oscillator to the chaotic portion of the circuit(e.g., resistor 225 in FIG. 2A), and is superimposed on the graph inFIG. 8 to illustrate the location of the strange attractor equilibriumpoints. As explained previously, the intersection of load line 850 withthe current-voltage characteristic curve determines the location of thestrange attractor equilibrium points. Depending on the orientations ofthese curves, the transmitting circuit can be made to operate in asingle-scroll or double-scroll attractor mode. See, e.g., FIG. 1D. Othermultiple-scroll modes, such as triple-scroll and the like are alsopossible. Single-scroll attractors orbit about the points where loadline 850 intersects Gb lines 860 a or 860 c. Double-scroll attractorstransition through the Ga line 860 b and move to or from Gb lines 860 aand 860 c. A triple-scroll attractor can orbit around the IVcharacteristic curve origin at the center of the Ga 860 b line.

Various second-generation embodiments of the invention modulate anonlinear circuit element by shifting the equilibrium point of thestrange attractor (for single-scroll circuits) or attractors (formulti-scroll circuits). This can be done by changing slopes Ga, Gb, orboth, of characteristic curve 860, which can be accomplished by changingvarious resistances within the nonlinear circuit element. Consequently,one or more slopes are changed in accordance with an information signalin order to modulate the position of the strange attractor (seegenerally FIG. 19D). In various first-generation embodiments, load line850 can be modulated. This load line can be used to shift theequilibrium point and change the operating point as shown in FIG. 1D(element 150). Control of the load line, however, affects the tankcircuit reactive load, which impacts modulation performance.

For a single-scroll attractor, the circuit will oscillate in only onequadrant of the characteristic curve (see FIG. 19C), and thus only onepiece (e.g., either 860 a or 860 c, depending on which quadrant thecircuit is oscillating) must be shifted to accomplish this modulation.For circuits operating in a double-scroll attractor mode, both pieces860 a and 860 c must be simultaneously shifted to cause both attractorsto shift along the curve, since at any particular point in time itcannot be predicted which quadrant the circuit will be operating in. Insecond-generation systems, one can control the transition of adouble-scroll or single-scroll attractor by modulating either Gb 860 aor Gb 860 c to cause the attractor to change from the upper scrollregion to the lower scroll region or vice versa. In this case, only oneGb line needs to be changed. This can cause a single scroll attractor tochange quadrants and remain in the other quadrant until its Gb line ismodulated to cause a transaction.

FIG. 6B shows a so-called “Caltech” diode, which produces acharacteristic curve similar to that of the Kennedy diode. Caltech diode650 includes two diodes 652 and 655 connected to a four-resistor networkcomprising resistors 651, 653, 654, and 658, each of which is coupled toeither a ground, the negative voltage supply, or the positive voltagesupply. Additionally, an op amp 657 and two resistors 656 and 659 arecoupled in the circuit. As explained below with reference to FIG. 7A,resistor 660, which is coupled between op amp 657 and ground, providesone possible mechanism for modulation in accordance with variousprinciples of the second-generation system. References herein to“Caltech diode” should be understood to refer to subcircuit 650 of FIG.6B and its many obvious variations. FIG. 6C shows yet another negativeresistance diode circuit 680 that can be used in a modulated transmitteraccording to various principles of the present invention. The novelcircuit shown in FIG. 6C includes four diodes 681 through 684 arrangedin pairs with opposing polarity across the terminals of capacitor 215through resistors 685 and 686, respectively. Op amp 689 and associatedresistors 688 and 690, along with resistor 687, perform functionssimilar to those of the Caltech diode shown in FIG. 6B. Referencesherein to “SAIC diode” should be understood to refer to circuit 680 ofFIG. 6C and its many obvious variations.

Turning now to FIG. 7A, one technique for modulating a nonlinear elementwill be described in detail, using the Caltech nonlinear diodeembodiment as the basis for discussion. (In FIG. 7A, elements withnumerals matching those in FIG. 6B are the same as those in FIG. 6B). Asshown in FIG. 7A, Caltech diode 650 is modified by adding a modulationcircuit 700 including switch 735 c and resistor 761 in parallel withresistor 660. An information signal 736 modulates switch 735 c andcauses resistor 761 to be switched into and out of the circuit. Whenswitch 735 c is open, a chaotic transmitter circuit incorporating diode650 oscillates chaotically around a single scroll or double scrollattractor. (The size of the modulating resistor controls the type ofscroll obtained). When switch 735 c is closed, resistor 761 changes theresistance of the nonlinear element in the transmitter, thus changingthe slopes Ga and Gb (see FIG. 8) and moving the position of the strangeattractor to a new equilibrium position (see FIGS. 19D and 19E). Thisnew equilibrium position causes a receiver synchronized to the previousequilibrium position to fall out of synchronization, thus transmitting abit of information.

In accordance with various second-generation embodiments, an informationsignal modulates a chaotic transmitter by changing a resistive part ofthe nonlinear element in the circuit, thus causing a near instantaneousshift (less than a quarter cycle of the tank circuit's fundamentalperiod) in strange attractor characteristics in the circuit. Otherimplementations that use transistors, field effect transistors, or otheractive devices can effect the modulation scheme described in thisparagraph assuming they implement the general current-voltagecharacteristic curve shown in FIG. 8. The nonlinear element can bemodulated using different resistive components as described with respectto the first-generation embodiments (e.g., optoisolator, analog switch,field effect transistor, relay, etc.) using a resistor in series with anideal switch and placing this combination in parallel with the nonlineardiode resistors. Other variations of modulating the resistive value areof course possible.

In FIG. 6A, for example, an additional resistor can be switched inparallel with resistor 608 to change the strange attractor trajectory ina Kennedy diode implementation. In FIG. 6C, for example, an additionalresistor can be switched in parallel with resistor 687 to change thestrange attractor trajectory. Other combinations of resistors can alsobe changed in the circuits to change the trajectory in any desiredmanner. In general, changing these resistances in the nonlinear elementof the chaotic transmitter circuit increases modulation bandwidth fromapproximately 10-15% of the tank circuit fundamental frequency to 200%of the tank circuit frequency when compared to modulating the reactivecomponents.

Many different resistive components of the circuits described above(e.g., resistors 651, 653, 654, 658, 659, 660, etc.) can be modulatedusing an ideal switching element to modify the trajectory in phase spaceof the chaotic strange attractor. Good results were obtained bycontrolling resistor 660 of FIG. 6B to simultaneously change the slopeof Ga and Gb as shown in FIG. 8. As shown in FIG. 7A, by switchingresistor 761 into and out of the circuit, the rotation of the strangeattractor phase plane changes as the slope of the current-voltagecharacteristic curve shown in FIG. 8 changes. This implementationchanges the slope of both Ga and Gb simultaneously and increases themodulation and demodulation rate (i.e., synchronization rate) at thevarious receivers described with reference to the first-generationsystems.

With reference to FIGS. 7A and 8, slope Ga is related to resistor 660(R1) by the equation:Ga=−1/R1and slope Gb is related to resistor 653 or 654 (R2) and the resistanceof the switch 735 c by the equation:Gb=−(R2−R1)/R1R2.

Referring again to FIG. 8, modulating resistor R1 shifts thecharacteristic curve of the transmitter by moving the slopes Ga and Gb.At the receiver, this characteristic curve is seen as a voltagedifference between the received signal and the signal generated by thereceiver (see, e.g., FIG. 4C). This voltage difference at the receiveris independent of the type of strange attractor (i.e., single ormultiple scroll) and results in an improvement over transmitters thatuse only single-scroll attractors. This modulation technique alsoincreases the available set of strange attractors, thus increasing thenumber of simultaneous transmitters that can operate in a givenenvironment and increasing noise immunity.

Returning again to FIG. 7A, frequency changes caused by modulating theinductor 230, capacitors 215 and 220, or resistor 225 cause zerocrossing voltages (when changing from an upper to a lower scroll or viceversa) that generally cannot be detected by the aforementionedreceivers. This inability to detect zero crossing (Ga region changes)voltage differences results in lost bits and therefore bit errors at thereceivers when using a double or multiple scroll attractor. Modulatingthe nonlinear diode current-voltage characteristic curve provides animprovement in detection within a Ga region that allows the use of anytype of attractor as the carrier for the modulating digital bit stream,rather than only a single scroll attractor. This modulation techniqueallows modulation to be applied at two times the tank circuit frequencycompared to 10-15% of the tank circuit that can generally be achieved bymodulating the other components of the transmitter.

In FIG. 7B, switching element 735 c in series with resistor 761 (whichtogether are in parallel across the nonlinear diode resistors) can beused to modulate components 660, 653, and 654 of FIG. 7A. The resistors653, 654, 660 can be individually modulated or modulated at the sametime to achieve different characteristic curve shapes. Elements 653,654, and 660 can be modulated singly to change slope Gb in the upperleft quadrant of FIG. 8 or Gb in the lower right quadrant of FIG. 8.They can be modulated as a pair to change both Gb slopes. Resistor 660can be modulated singly to cause Gb and Ga slopes on the characteristiccurve to change, or it can be modulated in conjunction with resistors653 and 654 to cause larger variations in the Gb slope. The Gb slope isrelated to the Ga slope in the Caltech diode of FIG. 7A as follows:Gb=Ga−1/R ₂ where R₂ is element 653 or 654.FIG. 7C summarizes the effect of resistor changes in nonlinear diode onslopes Gb and Ga. In FIG. 7C, slope element Gb (elements 860 a and 860 cin FIG. 8) and Ga (element 860 b of FIG. 8) are changed by varying theresistors for the Caltech and Kennedy diodes. The Ga and Gb slopes areshown down the left hand side of the page and the resistor values areshown across the top of the table. The slopes that are affected byvarying the resistors singly or in pairs is shown by an “x” in the box.It should be noted that several resistors affect all slopes and severaleffect only one or two slopes. Other variations beyond thosespecifically shown in FIG. 7C are possible. Changing the values ofresistors 651 and 658 (see FIG. 7A) changes the characteristic curvebreak points (see FIG. 8) by changing the breakdown/biasing points ofdiodes 652 and 655. The same technique can be used with any othernonlinear diode characteristic curve where the components that determinethe nonlinear diode current-voltage characteristic curve can bemodulated in a like manner to produce the characteristic curvemodulation.

The parallel resistors can be modulated over a wide range. To maintainchaotic behavior, characteristic curve 860 should intersect load line850 at three or more points, as shown in FIG. 8. The load line can beright on top of Ga 860 b and intersect the entire Ga line as well as thetwo breakpoints that lie on both the Gb and Ga. The closer the parallelcombination of switch 735 c, modulation resistor 761 and originalresistor 660 are to the original resistance 660 value, the less thecharacteristic curve is changed and the more difficult it is to detectthe modulation. The larger the combined resistive difference betweenmodulation resistor 761 and original resistor 660, the larger thevoltage difference generated at the receiver (i.e., the greater thecharacteristic curve is changed). Similar results can be obtained usingthe Kennedy nonlinear diode of FIG. 6A by modulating the variousresistors in the diode (e.g., resistors 604, 608, 607, and 601).

FIG. 7B shows generally how an information signal 736 can be used tomake and break a switch 735 c (or other switch-like device) to modulatea resistance 720 in a nonlinear circuit element according to varioussecond-generation embodiments. The resistance modulation scheme of FIG.7B can be applied to any type of negative resistance nonlinear element,including the nonlinear diode circuits of FIGS. 6A, 6B, and 6C, andothers. In general, a resistive value 720 in the nonlinear element ismodulated across its terminals 705 and 710 by applying a secondresistive element 761 in accordance with an information signal, whichmay comprise for example an on-off keying type signal. It will beappreciated that multiple values and switches can be used to create amulti-key and multiple modulation techniques. For example, multipleresistors and switches can of course be used to create a “vocabulary” ofresistive values, each corresponding to a different equilibrium point onthe current-voltage characteristic curve, with corresponding modulationand demodulation.

In FIG. 7B, the modulation scheme can be used to individually modulateone resistor at a time or used to modulate multiple resistorssimultaneously to get a different strange attractor. This can be done inpairs or all resistors in FIGS. 7A, 6A, 6B, and 6C in any combination.This creates flexibility in strange attractor designs. It is alsopossible to modulate resistor 720 directly using a photoresistor or FETdevices' internal resistance, rather than switching a resistance.Moreover, other types of nonlinear circuit elements could be used inplace of a diode. Examples include gas breakdown tubes or operationalamplifiers.

FIG. 9A shows various modulation limits of the nonlinear diodecharacteristic curve used in a Chua's circuit. Starting with the loadline 970 defined by −1/R, it can be seen that the load line varies from−Vsat 950 to +Vsat 960. These two points are the extremes of thenonlinear diode operational amplifier's voltage swing and are related tothe power supply voltage (Vcc) by approximately (Vcc−0.712 volts) inactual practice.

Referring to both FIGS. 9A and 9B, the nonlinear diode curve regions(901, 966, and 902) are also limited by this maximum voltage point. Thecurrent is limited by the maximum current drive of the operationalamplifier and this limit can be exceeded depending on the slope of 901,902, or 966. In addition, to operate in a chaotic state, load line 970should intersect the characteristic curve in at least three points (901,966, and 902). If these conditions are met, the transmitter willoscillate chaotically because break points 971 and 940 will be crossedby the voltage in the tank circuit and unpredictable behavior will takeplace in the form of an unpredictable transition in the strangeattractor orbits. As explained in more detail herein, chaotic operationcan also occur through the use of linear circuit elements, and theinvention is not limited to the use of nonlinear circuits to effectchaotic operation.

Under the above conditions, the modulation limits for chaotic operationare defined by the line segments 905, 970, and 945, and line segments965, 966, and 945. The lower curve is bounded by the −1/R load line(970) and the points where the slope of the line defined the points 905and 945 go to zero (i.e., no longer have a negative slope). If thetransmitter slope (905 or 945) becomes positive then a receiver wouldhave difficulty resolving where it was operating on the receivers'characteristic curve since two points could have the same current (i.e.,one on line 905 and one on line 970).

The second characteristic curve boundary is defined by regions 925, 965,and 935 and occurs where the current produced by a given slope Ga or Gbexceeds the nonlinear diode's current capability, or where Gb (925) nolonger touches the load line curve (970), within the operationalamplifiers' saturation voltage regions 950 or 960, or where the slope Gb(925 or 935) is zero or positive.

FIG. 9B shows graphically how modulation in the transmitter and receiveroperates. Transmitter voltage V1 (903) in the nonlinear diode produces avoltage V2 (904) in the tank circuit across resistor R associated withload line 935. V1 is transmitted through a channel to a receiver.

The receiver is a decoupled version of the Chua's circuit (see, e.g.,FIG. 19A). The receiver's tank circuit is coupled to the channel througha resistor R that is equal to R at the transmitter with a load line 931.Therefore, current flows in accordance with V1 impressed on the loadline 931, which generates a current I and voltage V2 (906). V2 (906) iscoupled through a unity gain operational amplifier to the nonlinearportion of the receiver, which generates a current I equal to thecurrent across the resistor R from the channel to the tank circuit. Thiscurrent then produces a voltage V1 (907) on the receivers' nonlineardiode characteristic curve. This voltage V1 (907) can be compared withthe input voltage (904), which generates a voltage difference across asynchronizing resistor Rsync, (e.g., resistor 385 in FIG. 4C), thusgenerating a bit of information. When the transmitter is then switchedback to the fundamental characteristic curve equal to the receiver'scharacteristic curve, the receiver and transmitter are insynchronization and a zero voltage difference across Rsync 385 isproduced. The incoming voltage then equals the voltage generated in thereceiver. The voltage difference across Rsync is therefore zero.

FIG. 9C provides an analysis of the positive-going breakpoint of thenonlinear diode, representing the voltage points in FIG. 6B. Thenegative-going breakpoint can be set in a similar fashion. The negativeresistance 660 is generated by the operational amplifier 657 in theCaltech diode shown in FIG. 6B. The voltage at point 242 must overcomethe breakdown voltage of diode 655 and the voltage bias set up byvoltage V+ set by the voltage division network made up of resistors 658and 654. This ratio is defined as (V+)*{[654]/([658]+[654])}. The sum ofthe voltage of resistor 655 and (V+)*{[654]/([658]+[654])} sets thebreakpoint for the nonlinear diode.

There is another way to shape the characteristic curve and set thechaotic operation. These same resistors 658, 654, and 660 can also setthe breakpoint. The diode voltage breakpoint 655 can also be raised byadding diodes in series as shown in FIG. 6C. Changing 660, 653, and/or654 change the slope of the nonlinear load line. This same technique canbe applied with any nonlinear diode. These diodes will have a differentset of resistor values that have to be manipulated in a like fashion tothe Kennedy diode of FIG. 6A. Other types of circuit elements could beused to modulate the nonlinear elements, such as photoresistors, FETs orother types of transistors could be substituted for the resistors tovary the resistance of these elements.

The zero crossings of the transmitted and received signal are maintainedirrespective of the curve being used at the transmitter. However, thereis a difference voltage at all other points on the receiver curve whenthe transmitter is using the modulating curve.

The equivalent-resistance of the modulating resistor and the resistorbeing modulated is given as follows:Parallel resistance equivalent={(resistance ofswitch[735c]+[761])*([660])}/(resistance ofswitch[735c]+[660]+[761])  (Equation 2).

This equation can be used to set any desired modulation resistor size toachieve the voltage difference to meet a particular communicationssystems noise performance requirements.

FIG. 10 shows a field effect transistor 1001 placed across diodes 652and 655 to implement on-offkeying according to a second-generationembodiment. In this embodiment, FET 1001 implements on-offkeying, whichallows rapid switching but results in a detectable modulation frequencyin the voltage output of the Chua's circuit across C1 (255) or C2 (260)in FIG. 2B. However, it is an improvement over current practice, whichresults in a better detection at the receiver output across a comparatordetector. Reference will now be made to FIGS. 20A through 231, whichshow various data plots and displays depicting voltages, currents, andfrequencies in systems employing the principles of the invention.

Double-Scroll Attractors

FIG. 20A shows a voltage (V1) versus voltage (V2) versus time (T) plotof a chaotic signal (double scroll strange attractor) withoutmodulation. FIG. 20B shows the plot of FIG. 20A when modulated with aninformation signal according to various second-generation embodiments.Voltage V1 is taken at point 242 in a chaotic transmitter (voltageacross capacitor 215, see FIG. 7A), while voltage V2 is taken at point282 in the same transmitter (voltage across capacitor 220). FIG. 20Cshows a plot of voltage (V2) versus current (I3) in inductor 230 as afunction of time without modulation, while FIG. 20D shows the samequantities when modulated with an information signal. FIG. 20E shows aplot of voltage (V1) versus current (I3) without modulation, and FIG.20F shows the same quantities when modulated.

In FIGS. 20A, 20C, and 20E, a 5.6 KHz strange attractor withoutmodulation oscillates about two equilibrium points. FIGS. 20B, 20D, and20F show modulation at 1000 bps applied to the transmitter nonlineardiode slopes Ga and Gb. This illustrates how the strange attractorparameters change with time and how modulating the non-linear diodeaffects the behavior of the system as the Gb line crosses the load lineat various points caused by the Gb slope changing.

The modulated signals were modulated at 1000 bps with a bit stream of“1010”. The modulated signal transmits the same strange attractor for abinary “1” and different strange attractor for a binary “0”. Thetransmission of a binary “0” shows a different time varying signal whencompared with the modulated strange attractor signal. Notice theun-modulated strange attractor signal and the modulated strangeattractor signal in FIGS. 20A and 20B are the same from time t=0 to 1millisecond due to transmission of a binary “1”, (i.e. synchronizedcase) while from time t=2 to 3 milliseconds in FIGS. 20A and 20B thestrange attractors differ due to the transmission of binary “0”, (i.e.non-synchronized case). (The same holds true for FIGS. 20C and 20D, andfor FIGS. 20E and 20F.)

The strange attractor signals are correlated whenever a binary “1” istransmitted and uncorralated whenever a binary “0” is transmitted giventhe receiver is tuned to receive a binary “1”. At the receiver, thesynchronized strange attractor produces a voltage across thesynchronizing resistor equal to zero. This establishes the synchronizedstate and produces a series of pulses corresponding to theout-of-synchronization state that corresponds to the spirals of thestrange attractor projected onto the V1 axis.

FIGS. 21A through 21C show modulation of the characteristiccurrent-voltage curve for a nonlinear diode, and FIGS. 21D through 21Fshow corresponding voltage-current plots for the modulated transmittercorresponding to FIGS. 21A through 21C.

In FIG. 21A, resistor R1 has a value of 1200 ohms, resulting inintersections between the load line and diode curve at approximately −5and +5 volts. In FIG. 21B, resistor R1 has a value of 1210 ohms,resulting in a slightly smaller voltage point for the intersections(i.e., about −4.75 and +4.75 volts). In FIG. 21C, resistor R1 has avalue of 1220 ohms, resulting in yet a smaller voltage point for theintersections (i.e., about −4.5 and +4.5 volts, respectively). ResistorR1 affects the nonlinear diode slopes Ga and Gb as shown in thesefigures. As R1 increases, the slope of Ga and Gb decreases. As Ga and Gbchange, the break point moves, causing the voltage swing to increase ordecrease.

Voltage-current plots corresponding to the above modulations aredepicted in FIGS. 21D through 21F. FIG. 21D shows voltage V2 versuscurrent in the inductor I3 plotted against voltage V1 for the case whereR1=1200 ohms. FIG. 21E shows the same quantities for the case whereR1=1210 ohms. FIG. 21F shows the same quantities for the case whereR1=1220 ohms. As can be seen in FIGS. 21D through 21F, as the resistanceis increased, the strange attractor trajectories become “squashed.” AsR1 increases, the amplitude of the attractor decreases in V1, V2, and I3due to a voltage decrease. This is due to the shift in the equilibriumpoints around the load line defined by the resistor R between the tankcircuit and the nonlinear diode circuit. As R1 increases, the plotsrotate in a clockwise direction.

FIGS. 21G through 211 show frequency versus amplitude plotscorresponding to the modulated parameters of FIGS. 21A through 21C. Inother words, FIG. 21G shows the spectrum when R1=1200 ohms; FIG. 21Hshows the spectrum when R1=1210 ohms; and FIG. 21I shows the spectrumwhen R1=1220 ohms. As can be seen, the frequency response shifts with anincrease in R1. This frequency shift can be used to determine whichvalue of R1 is switched in the circuit. In addition, the magnitude ofthe right most frequency spike changes with a change in R1, and thisamplitude can be detected using a filter and frequency detector. When R1is set to 1200 ohms, 1210 ohms, and 1220 ohms, respectively, the FFTspike is at 6.6 kHz, 6.85 kHz, and 6.95 kHz as shown in these figures.

Single-Scroll Attractors

FIG. 22A shows a voltage (V1) versus voltage (V2) versus time (T) plotof a chaotic signal (single scroll strange attractor) withoutmodulation. FIG. 22B shows the plot of FIG. 22A when modulated with aninformation signal according to various second-generation embodiments.These voltages are taken at the same points as in FIGS. 21A through 21C.

FIG. 22C shows a plot of voltage (V2) versus current (I3) in inductor230 (FIG. 7A) as a function of time without modulation, while FIG. 22Dshows the same quantities when modulated with an information signal.FIG. 22E shows a plot of voltage (V1) versus current (I3) withoutmodulation, and FIG. 22F shows the same quantities when modulated.

These figures also show the effect of starting the system with a set ofinitial conditions that are out of phase with the strange attractor. Thesystem spirals into the single scroll strange attractor and then builds,to its chaotic state conditions. As in FIGS. 20B, 20D and 20F, thesignals in FIGS. 21B, 21D and 21F were modulated at 1000 bps with a bitstream of “1010”. As with the double scroll strange attractor, thestrange attractor signals are correlated whenever a binary “1” istransmitted and uncorrelated whenever a binary “0” is transmitted giventhe receiver is tuned to receive a binary “1”.

At the receiver, the synchronized strange attractor produces a voltageacross the synchronizing resistor equal to zero. This establishes thesynchronized state and produces a series of pulses equating to theout-of-synchronization state that correspond to the spirals of thestrange attractor projected onto the V1 axis. The single scrollattractor detection is simplified since there is no zero crossing thatcan cause a detection error as the modulating frequency is increased.

FIGS. 23A through 23C show nonlinear diode plots similar to those inFIGS. 21A through 21C, but for a single-scroll attractor, whereinresistor R1 is 1930 ohms, 1940 ohms, and 1950 ohms, respectively.Resistor R1 affects the nonlinear diode slopes Ga and Gb as shown. As R1increases, the slope of Ga and Gb decreases. As Ga and Gb change, thebreak point moves, causing the voltage swing to increase or decrease.

FIGS. 23D through 23F show corresponding phase plots for the differentresistive values in FIGS. 23A through 23C. These phase plots show thatas R1 increases, the amplitude of the attractor decreases in V1, V2, andI3 due to a voltage decrease. This is due to the shift in theequilibrium points around the load line defined by the resistor Rbetween the tank circuit and the nonlinear diode circuit. The phaseplots also show that as R1 increases, the plots rotate in a clockwisedirection.

FIGS. 23G through 231 show frequency versus amplitude plotscorresponding to the different resistive values in FIGS. 23A through23C. The plots show how the frequency response increases with anincrease in R1. One can use this shift in frequency to determine whichvalue of R1 is switched into the nonlinear diode circuit. In addition,the magnitude of the right most frequency spike changes with a change inR1 in the nonlinear diode circuit, and this amplitude change can bedetected using, for example, an amplitude detection circuit. At R1values of 1930, 1940, and 1950 ohms respectively, the FFT spike is at6.425 kHz, 6.375 kHz, and 6.350 kHz, respectively as shown.

To summarize, a receiver that receives a signal generated in accordancewith the above modulation techniques would see a variation of thecurrent-voltage characteristic curve of the incoming signal andtherefore a difference between the incoming signal and the signalgenerated by a matched chaotic circuit at the receiver.Second-generation modulation techniques change the trajectory of acomponent that varies the trajectory about the strange attractor. Thisshows up in the signal as an equivalent (or apparent) amplitudevariation defined by the voltage across component 215, 220 and thecurrent through the inductor 230 in FIG. 2A. This is different fromconventional phase shift keying where the signal phase is shifted. Inthis case, the trajectory of the strange attractor determines a currentthrough inductor 230, and the two voltages across capacitors 220 and 215are changed within phase space when modulation is applied to one of theresistors in the diodes shown in FIG. 6A or 6B using an ideal switch(analog, FET, or relay) to switch a resistor in parallel with anynonlinear diode resistor components (e.g., any of resistors 601, 603,604, 606, 607, 608 in FIG. 6A, or any of resistors 651, 653, 654, 658,659, or 660 in FIG. 6B), or any combinations thereof.

Shifting the chaotic oscillator strange attractor trajectory by changingthe current-voltage characteristic of the transmitter's nonlinear dioderesults in an ability to modulate the strange attractor trajectory atmodulation rates as high as approximately two times the tank circuitfundamental frequency (f_(LC)) (the Nyquist rate 1/(2f) given by thetank circuit components) and still recover the signal using any of thereceivers described in the first-generation embodiments. This is anincrease of the modulating data rate by approximately 200% over usingfirst-generation transmitter modulation techniques that change afrequency by modulating other reactive circuit components (e.g.,elements 230, 215, or 220 in FIG. 2A) or resistor 225.

Modulating Ga and Gb causes the receiver to synchronize at the Nyquistrate of 1/(2 f). The output pulses exhibits this rise time at thereceive output of a comparator or subtractor circuit. This technique canalso be applied to a tunnel diode nonlinear current voltagecharacteristic curve for radio frequency operation or to a laseramplifier configured as a nonlinear diode.

The Fast Fourier Transform (FFT) of the voltage across C1 (element 215)and C2 (element 220) showed a 30 dB higher attenuation of the modulationfrequency component which was lost in the chaotic signal noise floorwhen modulating the nonlinear diode using R1 (element 660 in FIG. 7A) byless than one ohm. The modulating frequency was therefore not detectableas a distinct frequency component since it was below the chaoticfrequency spectrum noise floor.

Modulating the current-voltage characteristic curve results in a randompolarity reversal of the attractors, and at the receiver demodulatedoutput where the detected data is recovered. This random reversal adds asignificant improvement in the detectors' ability to recover aninformation signal in secure communication environments since it helpsmask the information signal in what appears to be additional noise onthe channel.

A similar effect can be achieved by modulation components 653 and 654since only Gb is modified. It will be appreciated that modulation can beeffected by changing the breakpoints in order to shift equilibriumpositions on the characteristic curve.

C. Receiver Synchronization

The following description explains how the chaotic receiver synchronizeswith signals generated using the techniques and systems described above,and how the value of the synchronizing resistor Rsync can be selectedfor optimal performance in a chaotic transmitter implementation.

Referring back to the receiver embodiment of FIG. 4C, an analysis ofsynchronizing resistor Rsync (element 385) when the nonlinear diode ismodulated determines the maximum voltage difference with and withoutmodulation applied across that resistor. As Rsync increases in value,the voltage difference across it increases with modulation as τ(Tau)=1/(resistor 385×capacitor 355)=1/(Rsync×C₁). This is anexponential time decaying voltage function. It has been determined thatthe decay rate of the time function τ optimizes the synchronizing signaldifference as Rsync approaches zero ohms but the voltage differenceacross resistor 385 approaches zero. The upper synchronization voltagedifference between a synchronizing signal and a modulating signal occursat Rsync=∞ (infinity).

In practice, Rsync can be optimized. It was determined empirically thatthe maximum frequency that could be demodulated in a receiver was twotimes the tank circuit (element 361) resonant frequency f_(LC), since amodulation rate greater than 2 f_(LC) at capacitor 360 in the tankcircuit charges the capacitor and does not allow sufficient time for thecapacitor to discharge. Capacitors 360 and 355 then charge to one of theequilibrium points of the strange attractor. On the characteristiccurve, voltage V1 goes to a value equal to the crossing of the directcurrent load line (1/R) where it crosses Gb in the upper or lowerquadrant as shown in FIG. 9B (line 904). Since this is a limitingmodulation rate, then this point can be used to set the time constantfor the 1/(Rsync×C₁) time-constant. This permits a modulation rate of 2f_(LC) and allows the largest voltage difference across the Rsyncresistor while still obtaining a synchronization time that supportsoptimum detection of the amplitude and frequency difference of theincoming signal (i.e., optimizes the voltage difference across Rsyncversus synchronization time). One can reduce Rsync even further but thevoltage difference is then exponentially decreased across Rsync for boththe synchronized and modulated signal. Optimum synchronization (in anoiseless channel) for circuit design can therefore be obtained whenRsync is selected as follows:Rsync#(1/(2f_(LC)×C₁)) where C₁ is capacitor 355.

This is an optimum design point for Rsync in a chaotic receiver. Thefrequency f_(LC) can be determined by taking the fast Fourier transformof the transmitter or by applying the following formula for an RLC tankcircuit:Frequency=k(1/(3.41(L×C ₂)^(0.5)))=k(1/(3.41(L×[element 360])^(0.5)))where k is a constant ranging from 1 to approximately 2.5 depending onthe internal resistance of the inductor (element 348) and the resistor R(element 380).

This establishes the upper limit for Rsync. It has been determinedexperimentally that in the presence of noise, Rsync in the range of lessthan about 1 ohm to about 1000 ohms improves the signal-to-noise ratioof the receiver circuits in the first and second generation receivers byseveral (e.g. two or more) orders of magnitude. This Rsync couplingresistor value forces the nonlinear diode portion of the circuit tosynchronize faster. The synchronization time in a noisy channeldominates the system's noisy performance. Thus, Rsync can be adjusted tooptimize the energy per bit to the noise per Hertz of bandwidth. Thiscan be achieved at resistance values for Rsync from about 1 ohm or lessto Rsync=(1/(2f_(LC)×C₁), see FIG. 4C, depending on the noisecharacteristics of the channel.

D. Gain Control Amplifier

The chaotic receivers exemplified in FIGS. 4C, 4D, 4E, and 4F can beoptimized by adding an amplifier in the circuit. FIG. 11 shows a gaincontrol amplifier 1146 inserted into a receiving circuit that reduces oreliminates the need for an automatic gain control (AGC) amplifier on theinput at point 1191. As shown, gain control amplifier 1146 is insertedinto the circuit between elements 1145 and 1165. Resistors 1188 and 1189set the amplification. In one embodiment, the optimum gain is 2.4 dB to3 dB.

The amplifier will also operate with gains above 3 dB. However, the sizeof the gain should be adjusted to ensure that the signal is not clipped.This amplifier reduces the need for an automatic gain control (AGC)amplifier on the input of the chaotic receivers at point 1191. Withamplifier 1146 in the circuit, the incoming signal can be attenuated bymore than 10 dB and chaotic synchronization will still occur insubsystem 1126 to generate voltage V1 across capacitor 1155. Thisamplifier also seems to provide some noise immunity because of its bandpass filter characteristics and its ability to move the signal away fromthe breakpoint of the nonlinear diode element 1126.

The most general case for this amplifier is shown in FIG. 12, whichrepresents an improvement over the system shown in FIG. 5. In FIG. 12,amplifier 1201 is inserted between subsystems 1230 and 1235 to overcomesignal attenuation in the incoming channel. An extension of this is toadd an amplifier between every subsystem that performs the function of areceiver element, such as subsystem 1230 and subsystem 1235. Theelements in FIG. 12 can be roughly mapped to previously describedembodiments as follows: first subsystem 1200 corresponds to the tankcircuit in the transmitter; second subsystem 1205 corresponds to thenonlinear element in the transmitter; third subsystem 1230 correspondsto the tank circuit in the receiver; fourth subsystem 1235 correspondsto the nonlinear element in the receiver; and synchronizing subsystem1250 corresponds to the synchronizing resistor and associatedcomponents. The voltage across tank circuit capacitor 1160 in thereceiver's tank circuit is amplified prior to being applied to resistor1165.

The IV characteristic curve in FIG. 9B illustrates how this amplifierworks. The received signal is compared to the signal produced by thereceiver. If the received signal was produced on Gb slope 901 or 902 inthe transmitter and it is attenuated below the breakpoint of thereceiver 971, then it is being reproduced based on the slope of Ga 966.The voltage in the tank circuit must be amplified enough to raise itabove breakpoint 971. This is based on slope Gb and load line 972. Thisthen causes a voltage on the nonlinear diode Gb slope 925, or 931instead of Ga 966. If the new voltage is compared to the incomingsignal, there will be a constant difference between the two signals.This then brings the systems into synchronization, which can be detectedas a one or zero voltage by Various threshold detection circuitsdiscussed herein.

In summary, the amplifier reduces the need for a precise automatic gaincontrol in the receiver system of a chaotic communications systemdesigned from Chua's circuit or any other nonlinear set of decoupleddynamical nonlinear equations used as a receiver. The amplifier allowsthe input channel signal to be attenuated up to 10 dB or more. Thismeans that the automatic gain control does not have to precisely matchthe incoming signal. Moreover, the amplifier also acts as a noisecanceling circuit based on its bandwidth filtering characteristics.

E. Chaotic Signal Filtering Transmitter

Certain second-generation embodiments of the present invention alsoinclude filtering in both the transmitter and receiver circuits. If asignal is produced by the transmitter and filtered, then the receivercan still follow this signal. At the receiver, the receiver follows thefiltered signal and oscillates in such a manner that it actually addsthe missing frequency components back into the receiver generatedsignal. The signal generated in the receiver must then be filtered usinga like filter to match the received signal. If the signal is notfiltered at the transmitter, then it can be filtered at the receiver toreduce noise effects. This can be done at the front end of the receiveror after the receiver has reconstructed the signal. The followingparagraphs describe this process.

Turning to FIG. 13, transmitting circuit 1300 is similar to thetransmitter of FIG. 2A, but includes an amplifier 1308 that isolatesfilter 1309 from the chaotic subsystem 222. This prevents the filter'sinput impedance from interfering with the chaotic operation of thechaotic circuit's nonlinear resistor 284 or capacitor 215 (see FIG. 2A).Filter 1309 can be a low pass filter to eliminate the direct currentcomponent, or a bandpass filter to allow only the higher order chaoticsignal components to pass over a real channel such as a radio basebandwhere direct current components are blocked. In practice, the threereceiver designs described above with respect to the first-generationsystems can be synchronized with output signal 1343 from filter 1309.This is an improvement because it can be used to match a chaotictransmitter circuit to the bandwidth of an audio, video, radiofrequency, or light (e.g., laser) channel, where frequencies below somecutoff frequency cannot be passed in a real system.

Considering a bandpass filter, filter 1309 bandwidth in practice can becentered around the tank circuit frequency (e.g., ±20% of the tankcircuit frequency) or centered around the information rate and stillsynchronize at the receiver. All the modulation techniques of thefirst-generation and second-generation embodiments can be applied withthe transmitter filtering principles of FIG. 13. Another way to judgethe bandpass requirement is to look at the bandpass of the tank circuitand design a bandpass filter that is slightly wider than it byapproximately 5%. Optimum selection should, of course, be made on anapplication-dependent design basis.

FIGS. 14A and 14B show low pass and bandpass filter characteristics ofone embodiment. Points 1400, 1420, 1440 are the 3 dB points of thefilters. In actual practice, the bandwidth can be twice this width sincemodulation rates of two times the tank circuit frequency are possible.Point 1430 is the center frequency of the bandpass filter andcorresponds to the fundamental frequency of the tank circuit 231 of FIG.13.

In general, placing a bandpass or lowpass filter at the output of thechaotic transmitter matches the transmitter to audio, video, light, orradio frequency channel modulators/upconverters. It also reduces noisecomponents outside these bandwidths.

F. Receiver Filtering

As shown in FIGS. 15 through 17, chaotic receivers can also benefit fromnoise filters. FIG. 15 shows a receiver including noise filters 1593,1594 and 1597 to filter out noise components introduced by thetransmission channel according to a second-generation embodiment of theinvention. FIG. 16 shows a receiver including noise filters 1666, 1694and 1697 to filter out noise components introduced by the transmissionchannel according to a second-generation embodiment of the invention.FIG. 17 shows a receiver including filters 1793, 1766, 1794, 1797 and1721, wherein filter 1721 works with automatic gain control amplifier1746 to further reduce noise generated by subsystem 1726. The noisefilters can be lowpass filters or bandpass filters that match thetransmitter filter, or a combination of low pass and bandpass filters.

Referring first to FIG. 15, two bandpass filters 1593 and 1594 filterout noise components introduced by the channel. The detector is thenplaced across the buffered input signal after filter 1594 and receiverreconstructed signal after filter 1597. This reduces the noisecomponents and improves detection in a noisy environment. The bandwidthof these filters can be set just like the description of FIG. 13 andFIGS. 14A and 14B.

FIG. 16 provides another implementation of noise filters. In this case,the incoming signal 1691 is applied to subsystem 1661 without filtering.The filtering is done in the synchronizing resistor chain made up ofcomponents 1666, 1653, 1687 and 1694. The noise is filtered from theinput and also from the output through 1696 and 1697.

The foregoing techniques can be used in conjunction with the modulationscheme of FIGS. 7A and 7B to optimize the detection of the chaoticstrange attractor trajectory shift keying to improve a communicationsreceiver detection performance.

FIG. 17 shows a receiver including filters 1793, 1766, 1794, 1797, and1721. Filter 1721 can be designed to work with the automatic gaincontrol amplifier 1746 to further reduce noise generated in subsystem1761. This receiver filtering technique will work with the receiverdesigns described above with reference to the first-generation systems.Unity gain operational amplifier 1722 acts as an impedance buffer.

G. Application to Cellular Phone Systems

The principles of both the first-generation and second-generationsystems can be applied to cellular and non-cellular telephony in severaldifferent ways. First, they can be used in a baseband modem. Second,they can be applied directly to an intermediate frequency level modem.Third, they can be used at the radio frequency level. In a cellulartelephone application, for example, a large number of matched strangeattractor pairs could be created to transmit information, much like codedivision multiplex systems operate today. Thus, for example, each of aplurality of modulators (or transmitters) in a base station would havestrange attractor parameters matched to a corresponding one of aplurality of portable telephone demodulators (or receivers); eachcorresponding demodulator/receiver in the base station would also bematched to a corresponding transmitter/modulator in one of the pluralityof portable telephones. If implemented using digital signal processingtechniques, the matching can be implemented with a software changerather than requiring specially matched hardware pairs.

In a baseband modem implementation, a modem takes advantage of the largeembedded base of cellular telephony equipment and infrastructure. Inthis implementation, the cellular phone technology acts as the carriersystem. The advantage of this method is that chaotic synchronizationtechniques could be used to produce digital modems for use in thecurrent generation of cellular telephones. Cellular phones would takeadvantage of the chaotic synchronization to synchronize in low signal tonoise ratio environments. This would allow digital data to be passed attwo to three times the current data rate of current cellular systems.

Turning first to FIG. 18A, various first and second-generationembodiments discussed in this patent application will be applied asdescribed. A chaotic baseband modem 1801 includes a chaotic transmitter1801A and a chaotic receiver 1801B. These circuits are coupled to aninterface circuit 1803, which in turn is coupled to a cellular telephone1801 (which may comprise a conventional cellular telephone of varioustypes commercially used as of the filing date of this application). Inthis embodiment, the output of the interface circuit 1803 would be inthe audio frequency range; the chaotic modem would essentially beinterfaced as though the cellular phone was a radio system that acceptedan audio input. Cellular telephone 1802 communicates with a conventionaltelephone system through one or more radio frequency communication nodesor base stations in a conventional manner.

The chaotic transmitter modem frequencies would fall into the audiobaseband of a cellular phone system (e.g., 300-3000 Hz). An improvementin data rate would result from applying the M-ary signaling techniquesdiscussed in the M-ary section below. The signal-to-noise ratioimprovement over current state-of-the-art modems (frequency shiftkeying) will allow cellular phones to operate over a longer range thancurrent systems. In other respects, the current generation cellularphones would remain the same.

Turning to FIG. 18B, in an intermediate frequency implementation, achaotic modem could have its frequency adjusted to operate, for example,on a 1 MHz to 10 MHz intermediate frequency (IF) band, but the inventionis not limited to this exemplary IF range. In this family ofembodiments, a chaotic intermediate frequency modem 1807 comprises achaotic transmitter 1807A and a chaotic receiver 1807B, wherein thebandwidth of the intermediate frequency could be set to 1.2 MHz to 2 MHzto match the code division multiple access signals used in currenttechnology. The radio frequency operation of the cellular telephone 1808would remain the same as today's state-of-the-art systems. Thisembodiment would require some changes to the cellular telephone from itsexisting commercial implementation.

FIG. 18C shows a chaotic modem that operates at radio frequencies. Thesystem includes a chaotic modem 1813 including a chaotic RF transmitter1813A and a chaotic RF receiver 1813B, which is coupled to an RFinterface circuit 1814 and antenna 1815. In this system, the frequencyof the chaotic system is increased to the cellular telephone frequencyband and the chaotic transmitter and receivers discussed herein areoperated at radio frequencies. The components may change to radiofrequency components such as tunnel diodes for the non-linear diodeimplementation and radio frequency amplifiers. In other respects, thechaotic transmitting and receiving circuits remain the same as in theembodiments described herein. The specification for frequency use can beset to the code division multiple access standard of 1.2-2 MHz. Thismeans that the tank circuit bandpass filter would be set to this 3 dBbandpass and the radio frequency transmission will be filtered to reducefrequency components outside of the passband.

The embodiments described above can take advantage of thecharacteristics of chaotic synchronization signals to synchronize in lowsignal-to-noise ratio environments. Various tradeoffs trades include (a)reducing the transmitter while keeping the data rate constant; (b)reducing the number of cells for a given area; (c) increasing the numberof users in a given cell size; and (d) providing new digital services inthe same bandwidth as current cellular phone systems. The interfacecircuits can be constructed as needed to interface existing or futuretelephone systems using well-known techniques.

In the embodiments exemplified by FIG. 18C, the new chaotic radiofrequency modulation scheme could take advantage of chaoticsynchronization spread spectrum capability where many more chaoticsynchronization signals could be mixed over the radio frequency spectrumand decoded much like code division multiple access is applied today.The number of users would be determined by the number of orthogonalspreading codes (chaotic synchronization signals) possible and thefrequency division spacing characteristics of chaotic signals asdiscussed with respect to the first-generation embodiments. Chaoticsynchronization offers the advantage of simpler transmitters andreceivers as well as noise immunity unachievable in conventional stateof the art systems.

H. Dual Transmitter/Receiver Combinations

FIG. 24 shows a dual-transmitter configuration (1200, 1205) according toa second-generation embodiment of the invention. In FIG. 24, two systems1200 and 1205 are used to generate two strange attractor pairs (e.g.,two single or two double scroll pairs). The strange attractors can begenerated using any of the techniques described above with respect tothe first-generation or second-generation systems. These strangeattractors are buffered using two unity gain operational amplifiercircuits 1201 and 1202 to prevent the switches 1220 and 1210 fromloading the chaotic circuit systems 1200 and 1205. For the purposes ofexplaining the operation, the nonlinear diode subsections 251 and 1251will form the basis for tuning the two chaotic systems. Each transmitter1200 and 1205 can be implementing using an embodiment of FIG. 6A, FIG.6B, FIG. 6C, or other embodiments described herein.

The slopes Ga and Gb of the nonlinear diodes can be set to two differentvalues. Chaotic circuit 1205 can include components set to differentresistance values than equivalent components of the nonlinear diodes inchaotic system 1200. This leaves the tank circuit subsystem elements 200and capacitor 215 in both Chua's circuit systems with equal elements. Inthe non linear diodes the operational amplifier 223 and its biasingresistors 202 and 203 are also equal. The only difference between thetwo systems 1200 and 1205 is the slopes of the nonlinear diodes 251 and1251.

An ideal switching system 1265 including single pole double throwswitches (or similar devices) 1210, 1220, inverter 1206, and informationinput source 1206 provide a means to switch the transmitter between thetwo strange attractors. A summing circuit 1295 sums the switched outputat point 1208 to provide a transmitted signal based on the ratio ofresistors 1270 and 1280 to resistor 1290 as in any summing circuitimplementation. In this case, a 1:1 ratio is used. This signal has highfrequency components due to the switching action of 1265.

To remove these unwanted switching pulses, low pass filter 1310 is used.Its lowpass cutoff frequency is set to the highest frequency of thechannel roll off. An attenuator circuit 1360 is then added to adjust theoutput signal to match the signal to the channel. A system such as thatshown in FIG. 24 (i.e., using two switched transmitters) provides moreprecise control of the strange attractors when compared to switching asingle transmitter's nonlinear diode or other elements values. Using twoor more switched transmitters allows one to choose the position of thestrange attractors in phase space (double scroll, single scroll in upperquadrant or lower quadrant for digital signals). Using two or moreswitched transmitters provides faster switching time since the switchingtime is no longer dependent on the reactive components of thetransmitter to stabilize. The switched transmitter in combination with areceiver pair allows the strange attractors to be optimized for the bestdetection results at the receiver.

I. Dual Receiver Design Using Common Tank Circuit

FIG. 25 shows an embodiment including a dual receiver (1370 and 601)using a common tank circuit 361. This design is dependent on using adual transmitter, or a single transmitter where the nonlinear diodeand/or the capacitor 215 is varied and the tank circuit 200 remains thesame. Lowpass filter 393 reduces the noise component and is set to thechannel bandwidth. The unity gain operational amplifier 351 buffers theline and drives the multiple circuit elements. The common tank circuitelement consists of elements 380, 348, and 360. The dual receiverelements consist of subsystem 601 and 1370. Receiver subsystems 601 and1370 are tuned to the transmitters' two strange attractor nonlineardiodes 251 and 1251 (FIG. 24) plus the capacitors 215 as shown in FIG.24. In FIG. 25, synchronizing resistors 1430 and 1450 provide a means todetect the difference between the incoming waveform and the strangeattractor produced by the dual receiver elements. Since the strangeattractors are different than those of the other of the subsystems 601and 1370, the difference across the two resistors is either zero whenmatched or some voltage difference when they are not matched. Unity gainoperational amplifiers 1400, 1410, 1420, and 345 isolate thesynchronizing resistors and the non linear diode elements from the tankcircuit and input signal 286 to prevent feedback from causing the systemto detune. This dual receiver provides a precise method to detect two ormore strange attractors. The nonlinear portions 1370 and 601 can beduplicated to detect additional chaotic strange attractors.

A dual receiver design such as the one depicted in FIG. 25 reduces thenumber of parts required in a receiver when capacitor 215 or thenonlinear diode are modulated or when one switches between two or moretransmitter circuits with a common tank circuit in FIG. 24. The voltagedifference across the synchronizing resistors 1450 and 1430 can bemaximized for detection by amplitude detection circuits as discussedherein. This also provides a convenient way to detect frequencyvariations between the two circuits at points 1470 and 287. The voltagedifference between points 1450 and 1430 can be maximized for amplitudemodulated signal detection by maximizing the Ga and Gb differences. Thedual receiver circuit design can be extended to N receivers whereelements 345, 365, 355, and 350 are reproduced with the values of thetransmitter systems.

J. Subtractor Circuit

FIG. 26 shows a subtraction circuit block for detecting the voltagedifference across the synchronization resistors 1400 and 1450 in FIG. 25and to detect the voltage difference between points 1470 and 287 of FIG.25. It includes buffer unity gain operational amplifiers 1710 and 1720in subsystem 1930 which isolates the subtractor circuit from thereceiver circuits. This is followed by an attenuator circuit subsystem1940 consisting of elements 1730, 1750, and 1760 as well as elements1740, 1745, and 1770.

The attenuation circuit reduces the voltage across the synchronizationresistors to a level the subtraction subsystems 1950 can process withoutclipping the input signals prior to the subtraction. The subtractioncircuit makes mathematical detection possible in an analog system.

K. Absolute Value Circuit

FIG. 27 shows an absolute value circuit made from an operationalamplifier circuit. The absolute value circuit can be used to take theabsolute value of the input signal. This circuit can be used for adetector. This circuit executes the following equation:Vout=abs(Vin)

Subsystem 1965 is an inverting operational amplifier that inverts theinput 1880 and provides it as an output 1920 to one side of the fullwave rectifier subsystem 1965. The input 1880 to the inverter subsystem1960 is input 1885 of the full wave rectifier. The final output 2045 ofthe circuit is then further processed to recover the amplitude modulatedinformation signal shown in FIG. 24 input data 1206 to complete thetransmitter receiver pair. This absolute value circuit allows amplitudemodulation on a chaotic signal to be processed into theinformation-bearing signal of FIG. 24 input point 1206. This circuit canbe replaced by a squaring circuit that performs the absolute valuefunction.

L. Dual Receiver Synchronization Detector Circuit

FIG. 28 shows a dual receiver synchronization detector circuit. In thiscircuit the inputs to the circuit 2000, 2010 and 2005, 2015 are placedacross the synchronization resistors 1450 and 1430 respectively in FIG.25. Input 2000 is tied to point 1440 and 2010 is tied to point 1470. Thedetector input 2005 is tied to point 1415. The detector input 2015 istied to point 287. The output of each subtraction circuit is thensubtracted as follows: 2020-2030=2035. Then 2035 is passed through anabsolute value subsystem 2040 (FIG. 27). The output 2050 is thenprocessed through a moving average detector and the original modulatingsignal at point 1206 in FIG. 24 is recovered after squaring the signal.

This detector provides a significant improvement in detector output overa simple comparator. The detector subtracts noise off the input signalsince the noise across the two synchronizing resistors of FIG. 25 aresubtracted off.

M. Dual Receiver Sync Detector Using Absolute Values

FIG. 29 works like FIG. 28 except the output of the initial subtractions2020 and 2030 are processed by an absolute value circuit to provide theabsolute value at 2060 and 2070. These two numbers are then subtracted.The difference should have the maximum difference since the dualreceiver circuit in FIG. 25 alternates between the two strangeattractors and produce minimum and maximum values in step with themodulating signals. The output of the circuit 2080 is then processedusing a moving average and square law detector and the originalmodulating signal 1208 in FIG. 24 is recovered by the detector. Thisdetector provides a significant improvement in detector output over asimple comparator. The detector subtracts noise off the input signalsince the noise across the two synchronizing resistors of FIG. 25 aresubtracted off.

N. Single Detector Circuit

FIG. 30 shows a single circuit detection circuit element. This elementcan be used to process the voltage at points 287 and 1470 of FIG. 25.Good results were obtained when the strange attractors generated by thedual transmitter in FIG. 23 were single scroll attractors with oneattractor produced on the upper quadrant Gb and the other single scrollattractor produced on the Gb in the negative quadrant. This maximizesthe difference between the equilibrium points. The circuit is believedto work best when the transmitter is switched between two single scrollattractors in two different quadrants of the IV characteristic curve inFIG. 9A, although double scroll and single scrolls in the same quadrantwould work, one produced on the +Gb slope and the other on the −Gbslope.

O. Interacting to Communications Systems

FIG. 31 shows a chaotic system interface in a chaotic circuit thatallows one to tap off the chaotic signal without interfering with thechaotic circuit's operation. Unity gain operational amplifier 2300provides the isolation. Since the chaotic signal voltage and directcurrent offset at the input 2200 are not at the level a communicationsystem expects to see at the baseband input 2350 (e.g., microphoneinput, data port input, etc.) a direct current offset circuit made up ofa direct current power supply 2230 and an attenuator subsection 2480made up of one or more attenuator circuits are required.

The use of more than one attenuator is usually required since the inputto most communication systems is in the range of hundreds of millivoltsand the output of a chaotic circuit is in the range of volts. Thedynamic range of one attenuator is usually not adequate and thereforeseveral are required to match the input voltage level 2200 the requiredoutput voltage level 2350. In FIG. 31 there are three attenuators. Thefirst attenuator consists of elements 2210, 2240, and 2700. The secondattenuator consists of elements 2290, 2300, and 2310. The thirdattenuator consists of elements 2320, 2330, and 2340. These threeattenuators make up the attenuator subsection 2480. In addition toattenuators this subsection can also act as an amplifier section bychanging the ratio of the resistor elements 2210, and 2240 or 2290 and2300 or 2320 and 2330. This makes subsection 2480 a universalamplifier/attenuation element. The level shifter subsection 2470consists of a power supply 2230 and an interface resistor 2250. Thisallows one to add or subtract a direct current level from the inputchaotic signal 2200 to match a single ended baseband communicationssystem at the output 2350.

This may be important in modem applications since many existing radiosand cellular phones have single ended baseband input/outputrequirements. This is an improvement over current practice since iteliminates the need for a direct current filter and preserves the directcurrent characteristics at the receiver since a known level can then beadded or subtracted from the incoming signal. This technique can beapplied to an amplitude modulated communication system (e.g., audio,radio frequency, or laser) and to a frequency modulated radio system.

The direct current offset subsystem 2470 is an improvement over currentpractice since it eliminates the need for a direct current high passfilter and preserves the direct current characteristics of the signal atthe receiver where this known direct current voltage level can then beadded or subtracted from the incoming signal. The attenuator/amplifiersubsection 2480 is an improvement in practice in interfacing a chaoticsignal voltage to the baseband of an existing communication system. Thetransmitter baseband interface circuit can be used in both radio andcable systems.

FIG. 32 shows a chaotic receiver baseband interface circuit. The outputof a communications receiver system baseband can be applied to the inputof the interface circuit at point 2650. Normally this is a single endedoutput and the signal must be amplified from millivolt levels to thevoltage levels expected by the chaotic receiver circuit at input point2830. Following a unity gain operational amplifier 2660 the signal isamplified by a low noise high gain amplifier subsection 2840 which isfollowed by a unity gain operational amplifier 2770 for isolation fromthe final gain stage consisting of elements 2780, 2820 and 2870.Inserted into the final amplifier section is the direct current voltageoffset subsection 2850. The output of this circuit 2830 is the recoveredchaotic voltage with the direct current voltage offset reinserted. Theoverall system element is designated subsystem 2860 and becomes aninterface block for any chaotic receiver block.

The direct current offset subsystem 2850 is an improvement over currentpractice since it adds the direct current characteristics of the signalback into the receiver signal where this known direct current voltagelevel was subtracted at the transmitter and can be reinjected into thesignal for processing by the chaotic receiver. The attenuator/amplifiersubsection 2840 is an improvement in practice in interfacing a chaoticsignal voltage to the baseband of an existing communications system. Thereceiver baseband interface circuit can be used in both radio and cablesystems.

FIG. 33 shows a chaotic interface system 2490 and 2860 used to interfacea chaotic transmitter and receiver to an infrared amplitude modulatedsubsystem 3090. The infrared transmitter consists of elements 3000,3100, 3020, 3040, infrared diode 3050, transistor 3060, and 3070. Thesubsystem 2490 shifts the direct current voltage level of the inputsignal and attenuates it from volts of signal 2200 to millivolts 2350 ofsignal (FIG. 31). The infrared receiver is a self biased phototransistorpair 3080. The output 2650 is then applied to the chaotic interfacecircuit 2860 at interface point 2650. The infrared transmitter andreceiver pair is a band limited system. This embodiment shows that achaotic signal can be sent over a band limited amplitude modulatedsystem and the original signal recovered adequately in the chaoticreceiver to pass information bits at the channel baseband bandwidth. Thechaotic interface circuit subsystem 2860 (FIG. 32) then changes thechaotic signal at interface point 2650 from millivolts to volts atinterface point 2930 with the direct current voltage offset added backinto the system. One infrared system embodiment had a bandwidth of 30kilohertz and the chaotic signal was a 5.6 kilohertz strange attractor.Applications of this system include remote control units as well ascommunications systems. In addition to infrared systems, the inventiveprinciples can be applied to other amplitude modulated systems includingaudio, radio frequency, and light (e.g., light emitting diodes and lasersystems).

Band limited communications systems such as an infrared communicationssystem can support a chaotic signal modem provided the interfacecircuits 2490 and 2860 are used to buffer the chaotic signal. Theinterface blocks provide a communications system capable of transmittingand receiving a band limited chaotic signal.

FIG. 34 is an embodiment of the chaotic signal interface circuit to aradio baseband system 3120. The resistor element 3100 provides animpedance match to the input of the transmitter baseband 2350. The radiofrequency transmitter 3100 can be any radio transmitter at anyfrequency. The radio receiver 3110 is matched to the communicationstransmitter. The output of the receiver 3110 is applied across aresistor 3120. This produces a voltage at interface point 2650. Thechaotic receiver interface circuit 2860 then matches the receiver signalto the chaotic receiver circuit. Chaotic signals can be passed over bandlimited frequency modulated systems using the interface subsections 2490and 2860.

FIG. 35 shows a cable driver system to pass a chaotic signal down atwisted pair or coaxial cable system. The chaotic interface circuit 2490buffers the signal by providing attenuation and direct current voltageoffset. The signal is then sent to a balanced line driver subsystem2455. The balanced line driver prepares the chaotic signal to go over atwisted pair cable and matches the impedance of the cable to the drivercircuit. One possible cable driver is derived from an operationalamplifier book (SAMS “IC Op-Amp Cookbook”, Third Edition, Walter G.Jung, 86-60253 (1997), page 387 titled differential line driver). Thiscircuit allows chaotic signals to be sent over cable systems.

A chaotic interface subsystem 2490 provides an advancement in practiceover current interface circuits and allows existing differential linedrivers to be used for chaotic systems.

FIG. 36 shows one embodiment of a cable line receiver circuit derivedfrom the SAMS Operational Amplifier book (page 346) with a chaoticreceiver interface circuit 2860. The subsystem 2880 buffers the signalreceived from the differential input amplifier with high common moderejection 2880. The circuit is a combination inverting attenuator andscaling adder which rejects common-mode input components whileamplifying differential ones. This system allows chaotic signals to bematched to a transmission line such as standard twisted pair. Thechaotic interface circuit allows chaotic signals to be passed over atransmission line.

FIG. 37A shows an embodiment of modulating the transmitter using theresistor 660 from FIG. 6B and measuring the voltage difference acrosselement 1185 of FIG. 11 at the chaotic receiver. This embodimentillustrates that the transmitter and receiver pair can act as a linearanalog system in regions 2100. This means that amplitude modulation ofthe nonlinear diode is possible and the modulating signal can berecovered by examining the voltage difference across 1185 of FIG. 11. Inaddition, this figure shows that there are a number of values forelement 660 in FIG. 6B 2110, 2120, 2130, 2140, and 2150 that are usableas signal vectors or code words in digital systems. Each of theseresistor values generates a unique difference at the receiver when thereceiver is set to a fixed resistor value R1. The signal vectors can beselected such that a maximum ratio is obtained. This curve can be drawnfor different transmitter receiver pairs for different 660 values anddifferent parameters of the Chua's circuit R, L, C1, and C2. The outputof each of the receivers can be correlated to a specific code word. FIG.37A can also be generated for resistor 653 and 654 in parallel with 660.

A separate set of curves was also implemented for the Kennedy diode asshown in FIG. 6A when the transmitter element 608 was modulated and thereceiver signal difference 3215 across elements 1450 and 1430 (i.e.,Rsync) of FIG. 25 was measured and plotted as FIG. 37B. The curves 3220,3230, 3240, and 3250 are examples of the voltage difference across Rsync1450 and 1430 as resistor 228 of the transmitter is varied. The linearregions of the curves (e.g., 3255) can act as an analog signaldemodulation curve by varying element 228 in the transmitter andobtaining a linear signal output at the receiver. Note the chaoticregion goes from approximately point 3310 to 3320. Within this band ofresistance, there are N possible signal vectors where N is determined bythe minimum detectable voltage difference at the receiver for variousresistor values 228 at the transmitter.

In FIG. 37B, the minimum voltage difference is where the transmitter andreceiver have the same nonlinear diode characteristic curves 3270, 3280,3290, and 3300 (i.e., element 608 in the transmit and receive nonlineardiodes are the same). Depending on the noise in the channel the numberof usable vectors is reduced. For example, if point 3260 is the noisefloor then the vectors would have to be adjusted to insure a detectablevoltage difference based on the modulated value of resistor 608. Onecould also look at a fixed resistor value such as 75 ohms point 3330 atthe receiver and look for the voltage difference at the receiver 3315 asresistor 608 in the transmitter changes between the vectors to determinea detection threshold. For digital signals, this means that one canspace the vectors at one ohm increments and detect the voltagedifference to all other signal vectors. In the presence of noise one mayonly be able to get some portion of this number based on the voltagedifference trigger setting 3260 or 3315. This curve can be plotted forother circuit elements and expand the number of signal vectors.

FIG. 37C shows what happens when the transmitter capacitor 215 is variedin FIG. 6B and receiver capacitors 355 and 1490 in FIG. 25 are set tofixed values with the nonlinear diode characteristic curve set at afixed value. The voltage difference across Rsync 1430 and 1450 of FIG.25 is plotted against the transmitter capacitor 215 of FIG. 6B. Thissame curve can be plotted for every resistor value in FIG. 37B and asurface constructed showing where the optimum capacitor and resistorvalues are for signal vectors in a digital communications system. Thisincreases the number of signal vectors possible with a given tankcircuit and capacitor 215. In FIG. 37C, the curve generated by varyingcapacitor C1 is plotted as 3400. There are two linear regions of thecurve defined as 3410 and 3420. These regions can be used as analogmodulation regions in an analog communications system. The voltagedifferences in conjunction with FIG. 37B can be used to form sets ofcurves that can be used to build code words when each value of R1 (608)is used to generate a new set of C1 (215) curves.

In a transmitter receiver pair configured as a communications system,analog communication is possible by modulating R1 (608) or C1 (215) inthe nonlinear diode of the chaotic transmitter circuit and observing thevoltage difference across Rsync in the receiver circuit using the linearregions of the detection curves in FIGS. 37A, 37B, and 37C. In thechaotic transmitter and receiver pair a large number of signal vectorscan be constructed using the curves in FIGS. 37 A, 37B, and 37C.

P. Using Multiple Transmitters and Receivers

Various systems and techniques described above use a technique ofmodulating current-voltage characteristics of a circuit element toachieve enhanced data rates in a basic digital communications channel.This technique can approach the Nyquist rate in its signal modulation.This is a fundamental improvement over the 15-20% performance ofprevious chaotic communications systems and is the basis for furtherimprovement to increase the information-signaling rate well beyond theNyquist rate. This is based on two fundamental factors that have beendescribed herein. First, the chaotic trajectory phase shift keyingtechnique has been shown to provide a robust method of signaling nearthe Nyquist rate using modulated chaotic attractors, which can bedetected by matched receivers using a robust process based on thereceiver improvements that are part of the technique.

Second, the transmitter and receiver circuits have been shown to becapable of generating a very large diversity of chaotic attractormodulated waveforms that can be discriminated from each other using theimproved receiver techniques. This enables a coding scheme in which eachtransmitted bit (or Tbit) is coded to a digital word (a sequence ofinformation bits—or Ibits) that is uniquely associated with one ofseveral transmitter/receiver/attractor combinations implemented in amultiple transmitter/receiver system. Thus Tbits are transmitted asdescribed above at a rate near the Nyquist rate, thus staying within thelimits imposed by basic physics and information theory, while Ibits aretransmitted at a higher rate, which is a multiple of the Tbittransmission rate and can apparently exceed the fundamental limits ofinformation theory.

This technique is illustrated in FIG. 38 for the specific case in whichtwo Ibits 4000 are coded into each Tbit 4065. This case requires 4transmitters 4020, 4030, 4040, and 4050 and receivers 4090, 4100, 4110,and 4120 in matched combinations. Each receiver is capable of uniquelysynchronizing with the waveform of its matched transmitter, and will notsynchronize with the waveforms of the other transmitters. Thetransmitters and receivers could be designed with identical tankcircuits and if values of the Ga and Gb parameters are varied asdescribed in the chaotic trajectory phase shift keying technique (seeFIGS. 24 and 25). Alternately, entirely different attractor types couldbe used for each transmitter/receiver pair so long as thesynchronization advantages are retained. Unlike chaotic trajectory phaseshift keying as described elsewhere herein, this technique isimplemented using unmodulated chaotic circuits. Individual Tbitwaveforms are formed by switching between the unmodulated outputs of thetransmitters at the Tbit signaling rate. This has the same effect andthe same characteristics as the previous modulation technique but ismore efficient when several or a large number of transmitter/receiverpairs are to be implemented.

The received signal consists of a series of Tbits, each with thecharacteristic waveform transmitted by the transmitter that correspondsto its coded information content. The train of transmitted Tbitsconsists of a patched sequence of different waveforms, and is to beanalyzed by the receiver circuit to recover the original sequence ofTbits in which the information signal is encoded. The signal is fedsynchronously to all the receivers, which implement a demodulation anddetection process similar to that described herein. The receiver thatmatches the transmitter of a given Tbit will synchronize using the rapidsynchronization methods described herein, while the other receivers willnot. In each Tbit period the matched receiver is identified and thecoded word is extracted to recover the original information signal, orthe Ibits in the coded word.

FIG. 38 starts with element 4000 for example, the four signal codingvectors. The input information sequence 4005 controls the order in whichthe coded 2 bit sequence vectors are transmitted via a set switch 4010.This set switch then controls the order in which the transmitters arepassed through the channel via switch 4060. The Tbit sequence 4065 isthen injected into the RF transmitter 4070 using a chaotic interfacecircuit such as that of FIG. 31 (element 2490). The RF receiver 4080then receives the signal and processes it to the baseband 4105 where itis applied to a chaotic receiver interface circuit such as that shown inFIG. 38 and processed to match the receiver input impedance the chaoticreceivers 4090, 4100, 4110, and 4120. Each of the transmitters can be achaotic circuit (such as a Chua's circuit) tuned to a specific strangeattractor. After the chaotic receivers are detector circuits 4130, 4140,4150, and 4160. Each of the receivers can be a matched Chua's circuitreceiver as discussed herein. The detector outputs are sent to selectorswitch logic 4160 that declares the received vector 4170. The code isthen recorded in a record code buffer and recovered in a 2-bit sequencebuffer to be played out in a serial fashion as the recovered bitsequence 4200.

The example in FIG. 38 uses a coded word consisting of 2 Ibits. The 4combinations of Ibit values result in a total of 4 digital words. With 4transmitter/receiver pairs each word is assigned uniquely to one pair sothat a complete coding (transform) of Ibits to Tbits is accomplished.Assuming that the basic communication process for Tbits (which followsthe processes described previously) and that the process achieves a Tbitsignaling rate of S bits/second with a bit error rate of BER at Tbitenergy E_(b)/N₀, the word-coded process would achieve twice the trueinformation transmission rate at half the equivalent (per Ibit) E_(b)/N₀and the same BER. Thus the extension of the basic chaotic trajectoryphase shift keying technique doubles the information rate withoutfundamentally changing the communications process.

This enhancement process is generalized by observing that 2^(N)transmitter/receiver pairs could be used to provide a complete code fordigital words of N Ibits each. Thus a 2^(N) dimension system has thecapacity to increase information rate by a factor of N withoutfundamentally changing the structure and physical communicationsparameters of the chaotic trajectory phase shift keying system describedelsewhere herein. If N is large then information data rates (Ibitsignaling rates) can be achieved far beyond Shannon's limit forcommunications systems. Although there are other methods that exceedShannon's limit in transmission rate, this technique has the fundamentaladvantage that it does not require the deep transmitted signalmodulation that can limit the compatibility of other techniques withvarious types of communications media with their requirements forsharing communications spectra.

This technique may require the use of error correcting codes. Althoughincreasing information rate by a factor of N does not increase the basicbit error rate, each Tbit error results in N correlated Ibit errors,analogous to a frame error in other digital communications systems. Thisis a disadvantage that must be addressed by error coding. However, thetechnique has the desirable property that the error correction codeswill operate on Tbits, at their lower transmission rate, in aconventional way. No new error correction techniques will be required toachieve acceptable performance with this technique despite the very highequivalent Ibit transmission rate that can be achieved. Varioustechniques for interfacing the trajectory shift keying signals to realcommunications systems at audio, radio frequency and laser lightfrequencies are discussed herein. These techniques translate to basebandmodems that can be used in cable transmission, radio and laser systems.This technique can be applied to a wide variety of systems using analog,hybrid digital-analog or pure digital components for implementing thechaotic circuits. The structure of such a system is shown in FIG. 39.Thus a system that uses a smaller number of transmitter/receiver pairs(up to approximately 2⁸=256 pairs) could use low cost ASIC chips withanalog or analog and digital components for both transmitter andreceiver systems. Much higher data rate systems (exceeding 2¹²=4096pairs) could be implemented in custom CMOS or other integrated circuitchips to provide a system with very high performance.

FIG. 39 is the general case where a codebook is made up of “N-bit” codeblocks 5000 with 2^(N) Transmitters 5030 and receivers 5090. Thetransmitters are multiplexed by element 5040 and switched in based onthe input bit sequence 5105 by 5020 and 5060 giving “Tbits” 5065. Thesignal is then interfaced to the RF transmitter 5070. The RF receiver5080 is interfaced to the 2^(N) chaotic receivers 5090. The output ofthe chaotic receivers are then multiplexed 5100 and decoded using 5110.The 2^(N) vectors are then recorded 5120 and recovered 5130. Finally thedata is serialized into the recovered data stream 5140 to become therecovered bit stream 5140.

For purposes of reference, the following is an estimate of the enhancedperformance that can be achieved with this technique. Shannon's limitimplies a limit of 2 bits per second (BPS) per Hertz (Hz) of channelwidth. The basic chaotic trajectory phase shift keying technique couldoperate at a Tbit signaling efficiency of about 80% with errorcorrection coding reducing Tbit efficiency by about 60% to a netefficiency of 32%. This would yield a Tbit signaling rate of 0.64 BPS/Hzof channel width. Without the enhancement this system could transmit9600 BPS through a typical 15,000 Hz channel. A 2⁸ dimension systemwould increase this to 76,800 BPS through the channel, which is farbeyond the capabilities of other existing systems. A 2¹² dimensionsystem could increase this to 115,200 BPS. Performance at these levelswould allow a more redundant level of error coding (multiple techniquesor coding plus redundant transmissions) while maintaining highperformance. Using this technique information data rate is limited onlyby the physical ability to fabricate components with a high packingdensity of transmitter and receiver circuits, and by availability anddesign of switching components to implement the process at the chiplevel.

Combined with the basic techniques for modulating and demodulatingchaotic signals described herein, the multiple transmitter/receiverenhancement applies to all communications applications envisioned forthis technology. The technique comprises a breakthrough for applicationsthat demand high information (Ibit) data rates but have fundamentallimitations on physical or Tbit transmission rates. These includerestrictions on use of the communications spectrum and limitations dueto transmission effects in the communications medium itself. Thus veryhigh speed modems will be possible using this technique, so long as thebasic communication process supports the use of basic chaotic trajectoryphase shift keying at conventional Tbit signaling rates as describedelsewhere herein.

Although there are other methods that exceed Shannon's limit intransmission rate, this technique has the fundamental advantage that itdoes not require the deep transmitted signal modulation that can limitthe compatibility of other techniques with various types ofcommunications media with their requirements for sharing communicationsspectra. The technique comprises a breakthrough for applications thatdemand high information (Ibit) data rates but have fundamentallimitations on physical or Tbit transmission rates. These includerestrictions on use of the communications spectrum and limitations dueto transmission effects in the communications medium itself.

Very high speed modems are possible using this technique, so long as thebasic communication process supports the use of basic chaotic trajectoryphase shift keying at conventional Tbit signaling rates as describedelsewhere herein. The same technique can be applied to the Gb-onlytransmitter and receiver pair discussed below by modulating theequilibrium points (e.g., by shifting to M-ary equilibrium points).

Q. Digital Version with Cube-Law Nonlinear Component

Various chaotic trajectory phase shift keying techniques describedherein use a chaotic oscillator circuit that has been modified inseveral respects to improve performance, but which retains the nonlineardiode component (for example, element 650 in FIG. 6B) in the nonlineardiode circuitry. When an all-digital implementation of the chaotictrajectory phase shift keying techniques is desired, however thiscomponent can be implemented more efficiently by other means. Onedigital implementation of chaotic trajectory phase shift keying can usea block diagram component using the following equation to model thecurrent vs. voltage characteristic of the nonlinear diode:I=−aV−bV ³where I is the current through the nonlinear diode and V is the voltageacross the diode. The constants a and b are calculated from the G_(a),G_(b), and B_(P) characteristics of the nonlinear diode circuit. Theconstants are constrained by two conditions necessary for the componentto produce the chaotic modulations necessary to implement thecommunications technique. These conditions are as follows:Condition 1: the slope of the equation at 1=0, V=0 must be less than −Gaso that the curve passes through the load line and will produce chaoticmodulations, producing the constraint that a>G_(a).Condition 2: the maximum usable value of V is defined by the point atwhich the slope of the I vs. V curve reverses sign and becomes unusable,producing the constraint that the maximum desired voltage V_(max) is asfollows:|V _(max)|≦(−a/3b)^(1/2).The third requirement that the curve passes through the origin at V=0and I=0 is automatically satisfied by this form of the equation.

The constants a and b can be calculated by imposing further constraintson the shape of the curve. Good performance can be obtained by using thefollowing procedure: (1) set V_(max) at the actual maximum voltageconsistent with the circuitry or other components used to fabricate thechaos producing circuit, and (2) constrain the curve to pass through aparticular point (V*,I*). These produce enough information to solve forthe constants. As an example, if the curve is constrained to passthrough the breakpoint in the nonlinear diode (B_(p),−G_(a)B_(p)) thenthe coefficients can be solved as follows: $\begin{matrix}{{Let}\text{:}} & {{r = {{\left( {B_{p}/V_{\max}} \right)\quad{and}\quad r} \leq 1}};} \\{{Then}\text{:}} & {{a = {3\quad{G_{a}/\left( {3 - r^{2}} \right)}}};}\end{matrix}$ b = −(G_(a)/V_(max)²)(3 − r²).Other solutions can be derived by changing the particular point (V*,I*)through which the curve is constrained to pass.

In a digital implementation of the equation it could be calculated asI=−V*(a+b*V*V), requiring the floating-point calculation of threemultiplications, one addition, and one sign change.

This nonlinear diode simulation technique allows for a significantlysmaller calculation workload than the nonlinear diode as modeledaccording to the Chua or Kennedy circuits. This leads to a moreefficient digital algorithm for implementing the component in a system,particularly when a multiple transmitter/receiver system necessitatesthe calculation of the component characteristic for a very large numberof different components.

R. Gb-Only Receiver

According to certain embodiments, the inventive principles can beemployed without using a strictly “non-linear” circuit element. Forexample, the receiver non-linear diode can be replaced with twofunctions that represent only the Gb slopes 5200 and 5210 as shown inFIG. 40. This receiver type will be denoted a Gb-only receiver. Onepossible implementation for such a receiver is shown in FIG. 41. Theslopes are represented by two equations as follows:Upper Scroll: I=Gb*V+Ga*Vbp−Gb*VbpLower Scroll: I=Gb*V−Ga*Vbp+Gb*VbpWhere the voltage at the zero current intercept is:V(upper)=+VBP(Ga/Gb)−VBP=5240V(lower)=−VBP(Ga/Gb)+VBP=5250

In the Gb-only receiver the Ga term is in the Gb slope equation as shownin the following equations:Gb=(R2−R1)/(R1*R2)=Ga−(1/R2) (see FIG. 6B)where R2 is elements 207 and/or 231 and R1 is elements 204 and 5360 ofFIG. 41. Only the Ga slope created by the nonlinear element is taken outof the receiver element.

In the receiver, this embodiment eliminates the breakpoints 5220 and5230 (see FIG. 40). This is important because noise when summed with thesignal can cause the incoming signal voltage to hit the breakpoint andswitch momentarily to the other scrolls. This can cause bit errors inthe recovered signal due to voltage spikes generated in the now-removedGa region (FIG. 40). Note the load line 5270 is maintained with respectto the transmitter implementation and the breakpoint voltage and currentpoints 5220 and 5230 are also the same as the nonlinear diode of thetransmitter.

This receiver embodiment allows operation in noisier channels whereother receivers would give bit errors when points 5220 and/or 5230 werehit due to channel noise. This receiver implementation also eliminatesthe problem associated with the signal being attenuated in the channel.In a diode receiver implementation described previously, when thebreakpoint voltage point is reached at either 5220 or 5230 a bit errorcan be induced in the receiver because of the slope change and theattractor transitioning to the other scroll region. With the slope Garemoved (compare FIG. 8), the strange attractor plus noise never causesa transition between scroll regions and the incoming signal is justattenuated towards the origin 5260 (i.e., the system never changesregions). If a lower scroll attractor was received then it is in theright half of the V plane and can be distinguished from the upper scrollattractor, which is in the −V plane.

In FIG. 41, a voltage difference is set up across the synchronizingresistors 1430 and 1450 in FIG. 41 equal to the current differenceassociated with the slope Gb. If a fixed length chaotic vector is sentand analog to digitally converted at the period of the information bits(data rate) of the transmitted signal, the system can distinguishbetween an upper and a lower scroll attractor. The upper scrollattractor will be in the negative portion of the V plane and the lowerscroll attractor will be in the positive V plane. This technique oflinearizing the receiver can be applied to any chaotic system that usesa nonlinear equation set (for example the Lorenz equation set).Controlling the transmitter in such a manner that it never hits abreakpoint ignores the problem of the receiver's being stimulated in thepresence of noise to hit the breakpoint. For optimum operations, two ormore single scroll attractor pairs in the same region (upper or lower)can be employed.

The technique discussed herein to transmit chaotic vectors (i.e., burstsof chaotic signal that do not cross the breakpoint voltage) when used inconjunction with a Gb-only receiver allows chaotic receivers of the typeshown in, e.g., FIGS. 41, 42, 45A, and 45B to operate in noisy channels.These Gb receiver embodiments can also be used with a transmitter thatuses feedback to stabilize the chaotic transmit signals to insure thechaotic signal never crosses the breakpoint voltage at the transmitter(i.e., stop a single scroll attractor). In a noisy system the Gbreceiver has no breakpoint voltage 5220 and 5230 (FIG. 40). For an upperscroll attractor, the noise must drive the system into the oppositeV-plane before bit errors are generated. This means that the systemgains the breakpoint voltage difference in receiver performance. Inaddition, the signal can be attenuated to nearly zero volts and still bedetected as a voltage difference across element 1430 and 1450 in FIGS.41, 45A, and 45B.

One detector using an analog-to-digital (sample and hold circuit)converter embodiment is shown in FIG. 42. In FIG. 42, the receiver isessentially the same as the dual receiver implementation shown in FIG.25. The change is that the nonlinear diode has been replaced by dual Gbreceiver only implementing elements 5340 and 5350 (i.e., one Gb in theupper plane and one in the lower plane of the IV characteristic curve asshown in FIG. 40, where the Gb values can be different depending on thestrange attractor pairs transmitted). The voltage sources 5310 and 5330represent the voltage crossings where the current flow is equal to zeroas shown in the equations above for the voltage crossings. The voltagesources are shown in FIG. 40 as points 5240 and 5250. The negativeresistor implementation represented by elements 202, 203, 223 and 204 aswell as 5360 remain the same. The resistor elements 207 and 231represent the resistance of the Gb portion of the nonlinear diode. Eachhalf of the circuit represents the upper and lower portions of the Gbnegative slope respectively 5340 and 5350. To detect the change inattractors, the detectors previously described in this document can beused.

In addition, a sample and hold circuit can be used to sample the signalat a multiple of the data rate to obtain a plot of the voltage at 381compared to points 1470 and 287. These plot the tank circuit voltage 361against the Gb receiver voltages 1470 and 287. If one has an upperscroll, then the voltage is in the negative portion of the V-plane andif it is a lower scroll attractor then it is in the +V-plane. Itrequires several volts of noise along the V axis of FIG. 40 to cause theV1 287 or 1470 voltages to change signs. This is a very noise immunereceiver implementation. This detector circuit will also detect twosingle scroll attractors, even if they are both in the same region(i.e., upper or lower scroll region).

FIG. 42 shows the sample and hold circuit detector (analog-to-digitalconverter) connected at points 1470, 287, and 281. A matrix of thevoltages 5360, 5370, and 5380 can be generated as shown to distinguishbetween the two attractor pairs. Another detection technique is tosample at points 1470 and 287 and correlate to a stored signal to make adetection decision. Since chaotic attractors can produce uncorrelatedsamples, the same receiver circuit can be used to detect two or moreuncorrelated signals. These signals can then be decoded using acorrelation circuit attached to points 381, 287, and 1470. The tankcircuit 361 tends to remove noise and provide an excellent detectiontechnique. It gives a positive means to detect the strange attractorvector pairs in noisy channels. By using a Gb only receiver the voltageat point 381 and 287 and 1470 never hit a breakpoint even in thepresence of high noise levels in the channel. This means the matrix inFIG. 42 can provide an accurate means of detection using single ormultiple samples of the signals until the noise drives the signal intothe opposite V-plane. This provides a noise improvement over othersystems.

In the Gb receiver implementation, the Gb receiver allows a widerdynamic range in the received signal level. The Gb receiver embodimentallows operation in higher noise levels by a factor of the breakpointvoltage since the voltage never changes regions unless the zero voltageline 5260 is crossed. The Gb receiver embodiment also simplifies thereceiver design since no diodes are required in the receiver. It alsoallows uncorrelated chaotic signal vectors to be received by the samedetector circuit and then separated by a correlator circuit.

The Gb-only receiver technique can be applied to any chaotic nonlinearsystem that has a nonlinear element or elements. The Gb receiver candetect double scroll or two single scroll attractors, even if they areboth in the same region (i.e., upper or lower scroll region), and canprovide an accurate means of detecting chaotic signals using single ormultiple sample and holds of the signal in noisy channels. Thisembodiment allows the signal to be attenuated in a channel by more thanan order of magnitude and still be detected by the Gb only receiver in anoiseless channel. In a conventional Chua's receiver using a nonlineardiode implementation the signal can be attenuated by no more thanapproximately 1-2 percent before the breakpoint is crossed and biterrors result in a noiseless channel.

In a noisy channel the Gb only receiver allows the signal to beattenuated by 50 percent or more and still detect the signal. Theattenuation of the signal does not affect noise performance. The energyper bit to the noise level is the only determinant when using a Gbreceiver. In some receivers, the noise floor is set by the breakpointvoltage based on the signal attenuation in the channel. In the Gb onlyreceiver, the transmitted chaotic signal can be attenuated to nearlyzero volts and still be detected as a voltage difference across element1430 1450 in FIGS. 41, 45A, and 45B. All filtering techniques in thisembodiment apply also to the Gb-only receiver.

Turning to the dual Gb-only receiver embodiment shown in FIG. 45A,negative resistor elements 5340 and 5350 of FIG. 41 are replaced withnegative resistors 5345 and 5355, and voltage sources 5310 and 5330 areplaced in series with resistors 5366 and 5365 respectively. The otheraspects of the circuit as similar to that shown in FIG. 41. Resistors5366 and 5365 are in series with operational amplifier 223 that convertsresistors 5366 and 5365 into negative resistors by changing thedirection of current flow. Resistors 5366 and 5365 are the parallelcombination of resistors 5366 and 204 and resistors 5365 and 5360respectively of FIG. 41. The resistors 5366 and 5365 are determined inthe same way as in the other embodiments described herein.

FIG. 45B shows yet another embodiment in which resistors 231 and 207 areinserted in parallel with negative resistor elements 5366 and 5365respectively. This makes up the resistive elements 5346 and 5356respectively. In this case, the parallel combinations of the resistorsare determined in the same way as FIG. 41. The difference between FIG.41 and FIG. 45A in this embodiment is that voltage sources 5310 and 5330are in series with the negative resistor chain made up of elements 223,5366, 5310 and 223, 5365, 5366 respectively. In this embodiment,elements 5345 and 5355 in FIG. 45A are replaced by elements 5346 and5356 in FIG. 45B. In FIGS. 41, 45A and FIG. 45B one starts by designinga standard Chau-based receiver as discussed above and translates the Gbslope into the negative Gb slope receiver to obtain noise immunity.These circuits provide 6-9 dB better performance than a standardnonlinear receiver with slope Ga present.

S. Gb-Only Transmitter

According to certain embodiments of the invention, chaotic transmittingcircuits exhibiting a linear (instead of nonlinear) current-voltagecharacteristic can be employed, wherein the slope of the curve isperturbed to transmit information. FIG. 43 shows a transmitter similarto FIG. 6C except the non-linear diode 680 has been replaced with the Gbslope element 5460. This transmitter produces single scroll attractorsin either the lower quadrant of the characteristic curve of FIG. 40(curve 5200) or the upper quadrant (curve 5200) depending on the valueof the voltage source 5420. Voltage source 5420 represents point 5240 or5250 in FIG. 40. The single scroll strange attractor orbits around theequilibrium point where load line 5270 crosses Gb line 5200 or 5210 inFIG. 40.

The resistive element 5410 has the same value as resistor 686 FIG. 6C.The resistor 204 is the same as resistor 204 in FIG. 6C. All other partsof the circuit are similar to those shown in FIG. 6C. The Gb slope canbe modulated by placing a switching element 235 c and a resistor 206across resistor 5410 and/or 206 as shown in FIG. 43. The voltage source5420 is varied as the slope is changed based on the same equations asthe equations for the Gb receiver discussed with reference to FIG. 41.The resistor 206 will be a different value depending on whether itmodulates resistor 5410 or 206. The input can be modulated by a digitaldata stream at point 236 (see FIG. 43).

FIG. 44 shows a dual Gb transmitter. It is similar to the dualtransmitter of FIG. 24 except the nonlinear diode subsections have beenreplaced by Gb-only negative resistance sections 5500 and 5510. Theupper Gb curve 5200 is implemented in subsection 5500 and the lower Gbcurve 5210 is implemented in subsection 5510 as shown in FIG. 40. Thevoltage sources 5520 and 5530 represent points 5240 and 5250 on FIG. 40respectively. The switching subsection 1265, summing amplifiersubsection 1295, filter 1310, and attenuator subsection 1360 perform thesame function as in FIG. 24 of switching between the upper and lower Gbsubsections 5540 and 5550 and buffering the signals 1350 before they areinjected into a baseband channel of a transceiver. The transmitterfiltering methods and circuit elements described previously areapplicable to the Gb-only transmitter embodiments and are incorporatedtherein.

T. Positive Slope Transmitters and Receivers

According to other embodiments of the present invention, positive-slopecircuit elements are substituted for various negative-slope elementsdescribed previously. Some of these embodiments are illustrated in FIGS.46 through 52.

FIG. 46 shows a positive slope (Gb+) current-voltage characteristiccurve for resistive elements 5600 and 5610 of FIGS. 47 and 48 (thecharacteristic curves are lines 5283 and 5290). Both lines have apositive slope as opposed to a negative Gb slope (compare with FIG. 40).

To design positive slope curves as shown in FIG. 46, the techniquesdiscussed for the chaotic systems discussed herein can be applied todetermine the maximum and minimum circuit components as shown in FIGS.19A through 19F. The difference is that the sign of Gb is changed from anegative sign to a positive sign. This shifts the −Gb slopes 5200 and5210 of FIG. 40 by ninety degrees. The −Gb slopes are rotated about thebreak points 5220 and 5230. The intersection of the +Gb slope with theload line 5270 defines asymptotically stable points to a specificvoltage level applied to the input to the receiver of FIG. 47.

Any incoming voltage in the receiver that is not matched to thesevoltage points generates a voltage difference across resistors 1450 and1430 in the embodiments of FIG. 47. In effect, this represents a high“Q” circuit to matched voltage levels established by the intersection oflines 5283 and 5290 with the load line 5270 at the input to the Gb+receiver of FIG. 47. The voltage cross points with the load line can bechanged by moving the breakpoint voltage points 5220 and 5230 in FIG. 46on the voltage axis thus changing the voltage sources +V 5240 and −V5250 as shown in FIG. 46. This can be done by changing voltage sources5310 and 5330 of FIG. 47.

Voltage sources 5310 and 5330 in the receiver of FIG. 47 are changed tomatch these voltage crossings in accordance with the equations for FIG.41 above, except the sign of Gb remains positive to create the positiveslope Gb+. M-ary voltage levels can be created where elements 1370 and601 are reproduced 2 ^(N) times as in the M-ary waveform discussionabove. In this case, the discriminating element is the equilibrium pointvoltage 5281 and 5282 formed by the high “Q” circuit consisting of tankcircuit 361 and resistive elements 5600 and 5610. This receiver system,when coupled with the transmitters of FIG. 49 and FIG. 50 and thedetector circuits discussed above, operates within 0.25 dB of the Eb/Nocurve for coherent optimum BPSK. However, it has characteristics thatdistinguish it from BPSK. First, the receiver does not need a phasecircuit. Second, the receiver still works through a filter even with 30%of a bit period time delay. Third, M-ary level operation is possiblewith the break point voltages adjusted. This system provides aself-synchronizing system while BPSK is not self-synchronizing.

FIG. 48 shows a Gb+ implementation of the dual receiver of FIG. 42 usinga positive Gb slope characteristic curve implementation as shown in FIG.46. The system can be implemented in a similar manner to that of FIG.42. The analog-to-digital converters perform the same function as theydo in FIG. 42 above. This implementation allows a digital system tosample and process the signal in a computer.

FIG. 49 shows a positive Gb slope transmitter. This embodiment of a Gb+transmitter can be modulated by switching resistor 5650 using a switch235 c and a resistor in parallel with resistor 5650. This changes theequilibrium point voltages 5281 and 5282 crossings on the load line ofFIG. 46 without changing the breakpoint position points 5220 and 5230.The Gb+ slope rotates about the old breakpoint. Another way to modulatethe transmitter is to change the voltage source 5675. This source movesthe zero current crossing of the characteristic curve without changingthe slope of the Gb+ element as shown in FIG. 51 (i.e., the breakpointvoltage is moved). Also, the new Gb+ line is parallel to the old Gb+line. At the same time, it changes the Gb+ characteristic curve crossingpoint of the load line in FIG. 46. This sets new equilibrium points.This allows M-ary equilibrium points to be set as discussed below.

FIG. 50 illustrates the use of a digital-to-analog converter toimplement a positive Gb+ slope transmitter. In this embodiment, the Gb+voltage levels are selected along the load line of FIG. 51 in accordancewith the voltage levels that can be generated by the digital to analogconverter (D/A) converter. This yields 2^(N) voltage levels where N isthe D/A converters' binary level capability. There are currently D/Aconverters capable of 2 to 16 levels. This embodiment allows levels tobe set to less than 0.01 volts or better.

The impulse response can also be applied through the D/A converter toexactly mimic a Gb+ transmitter. This D/A implementation confirms thatthe system looks like baseband BPSK with a direct current offset at thetransmitter. It is at the receiver that the new detector characteristicsare manifest. The receiver system becomes a high “Q” matched filter andsynchronizes on the equilibrium points in a binary or M-aryimplementation. The ability to operate with 2^(N) levels shows that thissystem is not just a BPSK receiver as discussed above. It hassignificant new characteristics (i.e., immunity to phase distortion andsignal delay).

The receiver of FIG. 47 is designed to have the same voltage crossingswhere there are 2^(N) receivers for each tank circuit. There can also bemultiple tank circuits with multiple loadlines 5270 to change theequilibrium points. The circuit elements can be selected on the basis ofvarious chaotic circuit implementations discussed herein with respect tothe first and second-generation systems. The ability to define theequilibrium points allows large numbers of high “Q” circuit elements tobe generated. The M-ary word sets can be based on equilibrium voltagelevels instead of waveforms as discussed herein. The reaction of thereceivers FIGS. 47 and 48 to a change in the incoming voltage levelallows the energy per bit to noise per hertz of bandwidth for thesenon-coherent receivers to come within 0.25 to 0.5 dB of the idealcoherent binary phase shift keying systems. This results in asignificant improvement in performance. This modulation technique willbe referred to as “chaotic impulse response modulation coding” and theM-ary coding will be referred to as “SAIC amplitude impulse coding.” Thereceiver acts like an impulse response circuit that quickly achieves theequilibrium level of the incoming waveform. In FIG. 50 the computingelement 5700 generates an information signal and loads the D/A converterwith an M-ary word 5710. The D/A converter 5720 then converts the M-aryword 5710 to an analog output 5730 where there are 2^(N) possibleamplitude levels 5730 based on the capability of the D/A converter 5720.The output is interfaced to the channel by the channel interface circuit5740. Element 5740 is the channel interface circuits discussed in FIGS.31 and 32. The output goes to the channels designated by 5750.

FIG. 51 shows one implementation of an M-ary SAIC coded signal. Thereare 1 to 2N level points 5271 through 5273. For example, the Gb+ slopes5250 and 5283 set equilibrium points 5271 and 5272. Equilibrium point5272 generates a voltage 5267 on the V-axis. This voltage is the voltagethat is transmitted over the radio channel and represents the SAICcoding of the modulating signal that changes the voltage source 5675 inFIG. 49 or is produced by the D/A circuit 5720 in FIG. 50. The load linecan also be varied to change the voltage crossings with the Gb+ curves.

FIG. 52 shows an example of a Gb+ slope being varied to change theequilibrium point; in this case, resistor 5650 FIG. 49 is varied withinthe chaotic stable range discussed above while breakpoint 5210 remainsat the same position. As the Gb+ slope is varied, the equilibrium pointis moved from 5272 to 5277 and the V-axis crossing is moved from 5285 to5286. This M-ary method can be implemented by increasing the number ofelements 5610 and 5600 in FIGS. 47 and 48 to 2, one for each level to bedetected. The shifting of the Gb+ characteristic curve and the load lineis done by moving the breakpoint voltage across the −Gb slope line 5200and 5210 of FIG. 52.

CONCLUSION

Thus has been described various systems, methods, and apparati formodulating and demodulating chaotic signals. It is apparent that many ofthe embodiments can be implemented using digital signal processingtechniques rather than analog circuits or discrete elements.Consequently, the claims should be interpreted to encompass suchcircuits and elements without limitation. No claim limitation appearingin the claims of this application should be interpreted to be in “meansplus function” format unless it explicitly recites “means for”performing a specified function.

1. A method of transmitting information, comprising the steps of: (1)generating a chaotic carrier signal that causes a voltage to oscillatechaotically about a first equilibrium point in a current-voltage phasespace of a circuit that exhibits a current-voltage characteristic curveon which the first equilibrium point falls by: generating a chaoticcarrier signal that oscillates about one of two equilibrium points inthe current-voltage phase space; and (2) changing, in response to aninformation signal, a non-reactive resistive value in the circuit andthereby causing the first equilibrium point to shift to a shifted firstequilibrium point in the current-voltage phase space by causing bothequilibrium points to shift in the current-voltage phase space.
 2. Themethod of claim 1, wherein step (2) comprises the step of switching anon-reactive resistive element in the circuit which changes a slope ofthe current-voltage characteristic curve for a circuit element.
 3. Themethod of claim 2, wherein step (2) comprises the step of switching aresistive element in a Kennedy diode circuit.
 4. The method of claim 2,wherein step (2) comprises the step of switching a resistive element ina Caltech diode circuit.
 5. The method of claim 2, wherein step (2)comprises the step of switching a resistive element in an SAIC diodecircuit.
 6. The method of claim 1, wherein step (2) comprises the stepof shorting at least two diodes arranged in opposite polarity.
 7. Themethod of claim 1, further comprising the steps of: (3) transmitting asignal resulting from the changed non-reactive resistive value through acommunication channel; (4) receiving the signal transmitted in step (3)in a receiver tuned to synchronize with the chaotic carrier signalgenerated in step (1); and (5) providing a demodulated output containingthe information signal by detecting periods of synchronization andnon-synchronization with the received signal.
 8. The method of claim 7,wherein: step (3) comprises the step of transmitting a single-scrollattractor chaotic signal; step (4) comprises the step of receiving thesingle-scroll attractor chaotic signal transmitted in step (3); and step(5) comprises the step of detecting periods of synchronization andnon-synchronization with the single-scroll attractor chaotic signal. 9.The method of claim 7, wherein: step (3) comprises the step oftransmitting a double-scroll attractor chaotic signal; step (4)comprises the step of receiving the double-scroll attractor chaoticsignal transmitted in step (3); and step (5) comprises the step ofdetecting periods of synchronization and non-synchronization with thedouble-scroll attractor chaotic signal.
 10. The method of claim 7,wherein: step (3) comprises the step of transmitting a triple-scrollattractor chaotic signal; step (4) comprises the step of receiving thetriple scroll attractor chaotic transmitted in step (3); and step (5)comprises the step of detecting periods of synchronization andnon-synchronization with the triple-scroll attractor chaotic signal. 11.The method of claim 1, wherein step (2) comprises the step of changing abreakpoint voltage of a piecewise linear response curve of the circuit.12. A chaotic transmitting circuit, comprising: an oscillator circuit; aresistor coupled to the oscillator circuit; a chaotic circuit, coupledto the oscillator circuit through the resistor, wherein the chaoticcircuit exhibits a current-voltage characteristic shape having a slopethat intersects a load line defined by the resistor and provides anequilibrium point about which a voltage oscillates chaotically; andmeans for changing the slope exhibited by the chaotic circuit inaccordance with an information signal that includes means for switchinga plurality of resistive values.
 13. The chaotic transmitting circuitaccording to claim 12, wherein the means for switching shifts a voltagebreakpoint on the current-voltage characteristic shape exhibited by thechaotic circuit.
 14. The chaotic transmitting circuit according to claim12, wherein the means for switching shifts a slope of a piecewise linearcurrent-voltage characteristic shape exhibited by the chaotic circuit.15. The chaotic transmitting circuit according to claim 12, wherein themeans for switching shifts two slopes of the current-voltagecharacteristic shape exhibited by the chaotic circuit.
 16. The chaotictransmitting circuit according to claim 12, wherein the chaotic circuitcomprises circuit elements having values selected so as to cause thechaotic transmitting circuit to oscillate about a single-scrollattractor.
 17. A system comprising a chaotic transmitting circuitaccording to claim 12 and further comprising a chaotic receiving circuitcomprising circuit components matched to synchronize with the chaotictransmitting circuit.
 18. A chaotic transmitting circuit, comprising: anoscillator circuit; a resistor coupled to the oscillator circuit; achaotic circuit coupled to the oscillator circuit through the resistor,wherein the chaotic circuit exhibits a current-voltage characteristicshape having a slope that intersects a load line defined by the resistorand provides an equilibrium point about which a voltage oscillateschaotically, wherein the chaotic circuit comprises a diode circuit thatexhibits a negative piecewise linear resistance; and a switch coupled tothe chaotic circuit, wherein the switch changes a nonreactive resistivevalue in the chaotic circuit in accordance with an information signaland thereby causes the first equilibrium point to shift to a shiftedfirst equilibrium point.
 19. The chaotic transmitting circuit of claim18, wherein the chaotic circuit element comprises: a first diodearranged in a forward polarity across the oscillator circuit through afirst resistor and coupled to a first voltage supply through a secondresistor; a second diode arranged in a reversed polarity across theoscillator circuit through a third resistor and coupled to a secondvoltage supply through a fourth resistor; and an op amp coupled to afirst group of three resistors, a first of which is coupled between anoutput of the op amp and a positive input terminal thereof; a second ofwhich is coupled between the output of the op amp and a negative inputterminal thereof; and a third of which is coupled between the negativeinput terminal and ground.
 20. The chaotic transmitting circuit of claim19, wherein the switch modifies a resistive value between the negativeinput terminal of the op amp and ground.
 21. The chaotic transmittingcircuit of claim 18, wherein the chaotic circuit element comprises: twoforward biased diodes coupled across the oscillator circuit through afirst resistor; two reverse biased diodes coupled across the oscillatorcircuit through a second resistor; and an op amp coupled across theoscillator circuit through a resistive feedback network.
 22. The chaotictransmitting circuit of claim 18, wherein the chaotic circuit elementcomprises two diodes arranged in opposite polarity across the oscillatorcircuit through corresponding resistors, wherein the switch shorts thetwo diodes in response to the information signal and causes the chaotictransmitting circuit to stop oscillating in a chaotic manner.
 23. Thechaotic transmitting circuit of claim 18, wherein the oscillator andchaotic circuit comprise circuit elements having values selected so asto cause the chaotic transmitting circuit to oscillate in asingle-scroll attractor mode.
 24. The chaotic transmitting circuit ofclaim 23, wherein the oscillator circuit comprises an inductance and afirst capacitance; wherein the chaotic circuit comprises a secondcapacitance; and wherein the values of the first capacitance, the secondcapacitance, the inductance, and the resistance are selected so as tocause the chaotic transmitting circuit to oscillate in a single-scrollattractor mode.
 25. The chaotic transmitting circuit of claim 18,wherein the oscillator and chaotic circuit comprise circuit elementshaving values selected so as to cause the chaotic transmitting circuitto oscillate in a double-scroll attractor mode.
 26. A nonlinear circuitelement for use in a chaotic transmitter, comprising: a first pair ofdiodes coupled in series and biased in a forward direction with respectto first and second circuit terminals; a second pair of diodes coupledin series and biased in a reverse direction with respect to the firstand second circuit terminals; a first resistor coupled between the firstpair of diodes and one of the circuit terminals; a second resistorcoupled between the second pair of diodes and one of the circuitterminals; an op amp coupled between the first and second circuitterminals through a resistive network; a fourth resistor coupled to theresistive network; and a switch that couples the fourth resistor intothe resistive network, thus changing a slope of the piecewise linearcurrent-voltage characteristic of the nonlinear circuit element inresponse to an information signal, wherein the first resistor, thesecond resistor, and the resistive network have values selected to biasthe nonlinear circuit element such that it exhibits a piecewise linearcurrent-voltage characteristic across the first and second terminals.27. A method of communicating between a portable telephone device and abase station, comprising the steps of: (1) generating an informationsignal at the portable telephone device; (2) modulating a chaoticcarrier signal with the information signal using a chaotic trajectoryshifting technique by changing a non-reactive resistive value in achaotic circuit element to cause a strange attractor trajectory shift;and (3) transmitting the chaotic trajectory shift-keyed signal generatedin step (2) to the base station.
 28. The method of claim 27, whereinstep (2) comprises the step of generating a chaotic carrier signal thatoscillates about two equilibrium points in a current-voltage phasespace, and further comprising the step of causing both equilibriumpoints to shift in the current-voltage phase space.
 29. The method ofclaim 27, wherein step (4) comprises the step of detecting periods ofsynchronization and non-synchronization between the signal the receivedchaotic trajectory shift-keyed signal generated and a locally-generatedchaotic signal using a circuit matched to a transmitter used to transmitin step (3).
 30. The method of claim 27, wherein step (2) comprises thesteps of: (a) modulating at a baseband frequency level; and (b)frequency translating the modulated baseband signal to a radio frequencyband.
 31. The method of claim 27, wherein step (2) comprises the stepsof: (a) modulating at an intermediate frequency band which falls betweena frequency band of the information signal and a radio frequency bandused by transmitting equipment; and (b) frequency translating themodulated intermediate frequency signal to the radio frequency band ofthe transmitting equipment.
 32. The method of claim 27, wherein step (2)comprises the steps of: (a) modulating the information signal directlyto a radio frequency band; and (b) directly transmitting the modulatedinformation signal in the radio frequency band.
 33. A chaotictransmitter, comprising: a first chaotic circuit that generates a firstchaotic signal having a first strange attractor trajectory; a secondchaotic circuit that generates a second chaotic signal having a secondstrange attractor trajectory different from that of the first strangeattractor trajectory; a switch coupled to the first and second chaoticcircuits, wherein the switch selects either the first chaotic signal orthe second chaotic signal in response to an information signal; alow-pass filter coupled to the output of the switch; and a summingcircuit coupled between the switch and the low-pass filter, wherein thesumming circuit sums the output from the switch.
 34. The chaotictransmitter of claim 33, wherein the first and second chaotic circuitseach generate a single-scroll strange attractor chaotic signal.
 35. Thechaotic transmitter of claim 33, wherein the first and second chaoticcircuits each generate a double-scroll strange attractor chaotic signal.36. A method of transmitting an information signal, comprising the stepsof: (1) generating a first chaotic signal comprising at least onestrange attractor that oscillates about a first equilibrium point; (2)generating a second chaotic signal comprising at least a second strangeattractor that oscillates about a second equilibrium point; (3) inresponse to the information signal, selecting an output of either thefirst chaotic signal or the second chaotic signal; (4) transmitting theselected output from step (3); and (5) filtering the output selected instep (3).
 37. The method of claim 36, wherein steps (1) and (2) eachcomprise the step of generating a single-scroll strange attractorchaotic signal.
 38. The method of claim 36, wherein steps (1) and (2)each comprise the step of generating a double-scroll strange attractorchaotic signal.
 39. The method of claim 1, wherein step (2) comprisesthe step of continuously varying the non-reactive resistive value over achaotic operating region in accordance with the information signal. 40.The apparatus of claim 12, wherein the means for changing continuouslyvaries a non-reactive resistance over a chaotic operating region inaccordance with the information signal.
 41. The apparatus of claim 18,wherein the switch continuously varies the non-reactive resistance overa chaotic operating region in accordance with the information signal.42. A method according to claim 1, wherein step (1) comprises the stepof using a digitally implemented nonlinear circuit having acurrent-voltage characteristic that satisfies the equation I=−aV−bV³,where a and b are constants.
 43. The chaotic transmitting circuit ofclaim 12, wherein the chaotic circuit comprises a digitally implementedcircuit having a current-voltage characteristic that satisfies theequation I=−aV−bV³, where a and b are constants.
 44. The chaotictransmitting circuit of claim 18, wherein the chaotic circuit comprisesa digitally implemented circuit having a current-voltage characteristicthat satisfies the equation I=−aV−bV³, where a and b are constants. 45.The method of claim 1, wherein step (1) comprises the step of using acircuit that exhibits a linear slope in one quadrant of thecurrent-voltage characteristic curve, and wherein step (2) comprises thestep of changing the linear slope in the one quadrant.
 46. The apparatusof claim 12, wherein the means for changing comprises a voltage sourceand a switch that shifts a slope in one quadrant of the current-voltagecharacteristic shape.
 47. The apparatus of claim 18, wherein the switchswitches a voltage source to shift to the shifted first equilibriumpoint.
 48. The method of claim 45, further comprising the step offiltering an output of the circuit to limit its frequency bandwidth. 49.The apparatus of claim 46, further comprising a filter coupled to anoutput of the chaotic circuit that limits a frequency bandwidth thereof.50. The apparatus of claim 47, further comprising a filter coupled to anoutput of the chaotic circuit that limits a frequency bandwidth thereof.51. The method of claim 1, wherein step (1) comprises the step of usinga circuit that exhibits a positive linear slope, and wherein step (2)comprises the step of changing the positive linear slope.
 52. Theapparatus of claim 12, wherein the chaotic circuit exhibits a positivelinear slope.
 53. The apparatus of claim 18, wherein the chaotic circuitexhibits a positive linear slope.
 54. A method of transmittinginformation, comprising the steps of: (1) generating a chaotic carriersignal characterized by a voltage that oscillates chaotically about afirst equilibrium point in a current-voltage plane, wherein the firstequilibrium point is defined by an intersection of a current-voltageload line having a first slope and a current-voltage slope line having asecond slope opposite in polarity to that the of the first slope; (2) inresponse to a time-varying information signal comprising an N-bitsymbol, selecting one of a plurality of 2^(N) equilibrium points definedby successive intersections of a plurality of current-voltage slopelines having slopes opposite to that of the load line and that intersectthe load line at different points; (3) shifting the first equilibriumpoint to the one selected equilibrium point such that the chaoticcarrier signal oscillates chaotically about the one selected equilibriumpoint by changing a nonreactive circuit value in a chaotic circuitcoupled to a resistor that defines the current-voltage load line; and(4) transmitting the chaotic carrier signal shifted in step (3).
 55. Themethod of claim 54, further comprising the steps of: (5) receiving thesignal transmitted in step (4); (6) determining which of the pluralityof equilibrium points corresponds to the signal received in step (5);and (7) on the basis of the determination in step (6), generating aninformation symbol.
 56. A method of interfacing a chaotic transmittingcircuit to a communications channel without using a frequency filter,comprising the steps of: (1) buffering an output of the chaotictransmitting circuit to isolate the chaotic transmitting circuit fromthe communications channel; (2) removing a direct current voltagecomponent from the buffered output obtained in step (1) by using adirect current power supply and an attenuator circuit; and (3) matchingthe amplitude and impedance of the signal obtained from step (2) to thecommunications channel.
 57. The method of claim 56, wherein step (3)comprises the step of using a balanced line driver to match theelectrical characteristics of a twisted pair wire communicationschannel.
 58. Apparatus for interfacing a chaotic transmitting circuit toa communications channel without using a frequency filter, comprising:an isolation circuit that buffers an output of the chaotic transmittingcircuit from the communications channel; a direct current power supplycoupled to the isolation circuit through a resistor, wherein the directcurrent power supply subtracts a direct current voltage from the outputof the isolation circuit; and an attenuator circuit, coupled to thedirect current power supply, wherein the attenuator circuit attenuates asignal present at the direct current power supply prior to beingintroduced into the communications channel, wherein the communicationschannel comprises a radio frequency channel.
 59. The apparatus of claim58, wherein the communications channel comprises a cable system.
 60. Theapparatus of claim 58, further comprising a balanced line driver thatmatches the electrical characteristics of the apparatus to a dualconductor cable.
 61. The method of claim 27, further comprising: (4) inthe base station, demodulating the transmitted signal to recover theinformation signal.